Similarity Symmetrical Equivalencies between ‘Demand’ – ‘Supply’ aspects in an...

Similarity Symmetrical Equivalencies

between ‘Demand’ – ‘Supply’ aspects in an interindustry system: Transformations, Weighted Multiplier Decomposition & Distributions.

XI International Conference on Input-Output Techniques New Delhi, India 27 November - 1 December 1995

Nikolaos K. Adamou Sage Graduate School, Albany NY, USA

ABSTRACT

This paper demonstrates the similarity symmetrical equivalencies between the demand driven production model and the supply driven allocation model in a balanced interindustry accounting system. Both models, based on the same data and the methodological assumption of linearity, provide the same solution for gross output. Each model provides different descriptive and some structural information. Both models contain similar matrices and therefore have the same determinants and eigenvalues. Eigenvectors are the same for all matrices in each one of the respective system. The descriptive and structural information of each modeling aspect is transformed among modelling aspects. The properly weighted gross output multiplier is the same in both aspects, and if decomposed appropriately, provides differently viewed diversified influences. A variety of descriptive and structural distributions extend the practical application of interindustry analysis.

1 INTRODUCTION

2 INTERINDUSTRY ACCOUNTING: THE COMMON BASE FOR THE DEMAND & SUPPLY ASPECTS IN INTERINDUSTRY MODELLING

3 DESCRIPTIVE COEFFICIENTS IN INTERINDUSTRY MODELLING 3.1 Column Distributions matrices

3.2 Row Distributions matrices 3.3 Two-dimensional Distribution Matrices of Intermediate Transactions, Final Demand and Primary Inputs 3.4 Two-dimensional Total Gross Output Distribution Matrices

4 INTERINDUSTRY ‘DEMAND’ & ‘SUPPLY’ MODELS AND EQUIVALENCE TRANSFORMATIONS 4.1 Same Solution from ‘Demand’ and ‘Supply’ Models

4.2 Quasi-Inverses 4.3 Multiplier Overestimation & Decomposition 4.4 Involved Determinants & Equivalent Similarity Transformations 4.5 Fundamental Interindustry Identity 4.6 Eigenvalues & Eigenvectors in the ‘Demand’ - ‘Supply’ Modelling

LIST OF TABLES & FIGURES

BIBLIOGRAPHY

Similarity Symmetrical Equivalencies between ‘Demand’ - ‘Supply’ aspects in an interindustry system 2

1 Introduction

The idea of sectoral interdependence among various economic activities was explicitly formulated first by Quesnay in his “Tableau Economique.” The Tableau was a chart showing the flow of production between social classes.1 Later, Marx used similar concepts in his reproduction process.2 One can also observe the concept of sectoral interdependence in Warlas.3 Leontief developed the interindustry model4 tractable for empirical research. The interdependence of the productive sectors provides the structure of the economy. Leontief’s model is based on the fact that production is the most important economic activity, and the other activities are related to and based upon the production process. Von Neuamann’s model of an expanding economy is an extension of Leontief’s model.5

Ghosh6 suggested that the same methodology might be used in allocation decisions. Yamada7 analyzed clearly the meaning of production and allocation interindustry modelling aspects, however this did not have any significant influence on western audiences. Augustinovics8 interpreted the structural information of both modelling aspects and provided proofs of their equivalence. Bulmer-Thomas9 discussed the differences between purchases and absorptions indicating that the fundamental assumption of input-output refers to the stability of the relationship between absorptions (not purchases) and gross output. Oosterhaven10 discussed the supply model in an interregional framework. Several authors11 have utilized 1 Maital S. (1972) “The Tableau economique as a simple Leontief Model: an amendment”. Quarterly Journal of Economics, 86 (3), pp. 504-507. 2 Morishima M. & F. Seton (1961) “Aggregation in Leontief matrices and the labour theory of value” Econometrica. 3 Morishima M. (1959) “A Reconsideration of the Warlas-Cassel-Leontief Model of General Equilibrium” in Arrow K. J., Karlin S. & Supposes

(ed.) Mathematical Methods in the Social Sciences. 4 Leontief W. (1936) “Quantitative Input and Output Relations in the Economic System of the United States” Review of Economics and Statistics,

XVIII. Leontief W. (1937) “Interrelation of Prices, Output, Savings, and Investment”, Review of Economics and Statistics, XIX. Leontief W. (1941) The Structure of American Economy, 1919-1929. Harvard University Press. 5 Gale (1960)The theory of linear models, chapter 9.3, pp. 310-316. McGraw Hill. 6 Ghosh, A. (1958) “Input-Output Approach in an Allocation System.” Economica 25, pp. 58-64. 7 Yamada I. (1961) Theory and Applications of Interindustry Analysis, Kinokunika Bookstore, Tokyo, Japan. 8 Augustinovics, Maria (1970) “Methods of international and intertemporal comparison of structure.” In Carter & Bródy (ed.) Contributions to

Input-Output Analysis, North Holland, Vol. I, Chapter 13, pp. 249-269. 9 Bulmer-Thomas, V. (1982) Input-Output Analysis in Developing Countries. p. 172, New York, Wiley. 10 Oosterhaven, J. (1981) Interregional Input-Output Analysis and Dutch Regional Policy Problems, pp. 138-155, Aldershot, England: Gover. 11 Marthur, P. N. (1970) “Introduction” Carter & Bródy (ed.) Contributions to Input-Output Analysis, North Holland. - Conway R. S. Jr., (1975) “A note on the stability of regional interindustry models” Journal of Regional Science, Vol. 15, pp. 67-72. - Gray, S. Lee, etc. (1979) “Measurement of Growth equalized employment multiplier effects: an empirical example.” Annals of Regional Science,

13 (3), pp. 68-75. - Clapp J. M. (1977) “The relationships among regional input-output intersectoral flows and rows-only analysis” International Regional Science

Review 2 pp. 79-89. - Giarratani F. (1976) “Application of an interindustry supply model to energy issues” Environment and Planning A, 8, pp. 447-454. - Giarratani F. (1980) “The scientific basis fir explanation in regional analysis” Papers of Regional Science Association 45, pp. 185-196. - Giarratani F. (1981) “A supply constrained interindustry model: Forecasting performance and an evaluation.” In W. Buhr and P. Friedrich (eds.)

Regional Development under Stagnation. Baden-Baden: Nomos Verlagsgesellshaft. - Socher C. F. (1981) “A Comparison of Input-Output Structure and Multipliers.” Environment and Planning A, Apr. 13 (4), pp. 497-509. - Bon, Ranko (1984) “Comparative Stability Analysis of Multiregional Input-Output Models: Column, Row, and Leontief-Strout Gravity

Coefficient Models.” Quarterly Journal of Economics, Nov., 99 (4), pp. 791-815. - Davis, H. C., and E. L. Salkin (1984) “Alternative Approaches to the estimation of economic impacts resulting from supply constraints.” Annals of

Regional Science, 18, pp. 25-34. - Cronin J. (1984) “Analytical assumptions and causal ordering in interindustry modeling” Southern Economic Journal 51, pp. 521-529. - Deman S. (1985) “Political economy of regional development: a review of theories” Indian Journal of Regional Science Vol. 24 pp. 192-200. - Primero, Elidoro P. (1985) “Effects of changing industrial structures and changes levels and composition of the final demand bill on the input

factor requirements of the economy: An application of input-output analysis.” The Philippine Economic Journal, XXIV (2&3), pp. 200-221. - Bon, Ranko (1986) “Comparative Stability Analysis of Demand-side and Supply-Side Input-Output Models.” International Journal of

Forecasting, 2, pp. 231-235. - Chen C. Y. (1986) “The optimal Adjustment of mineral supply disruptions” Journal of Policy Modeling, 8, pp. 199-221. - Chen C. Y., and A. Rose (1986) “The joint stability of input-output production and allocation coefficients.” Modelling and Simulation 17, pp.

251-255.

XI International Conference on Input-Output Techniques, New Delhi, India, 1995 3

the allocation technique in various applications. The use of the supply driven approach has generated lengthy discussion on its plausibility. The main problems are three: 1) the difficulty in interpreting fixed output12 coefficients the way fixed input coefficients are accepted, 2) the interpretation of the multiplier matrix (Ghoshian inverse) and 3) the problem of stability.

This paper attempts to clarify methodological aspects interrelating supply to the demand driven input-output model. An equivalence proof of both modelling approaches is provided. Transformations of all descriptive and structural information between both approaches are furnished. Detailed discussions on the weighted multiplier and its decomposition, as well as other distributions are examined for policy initiative exploration and simulation.

2 Interindustry Accounting: The Common Base for the Demand & Supply Aspects in Interindustry Modelling

Interindustry transaction flows is the starting point in interindustry modelling. Transaction flows provide information about sales and purchases of the industrial sectors indicating both production and allocation structure. Gross production is absorbed by industrial and final demand. The square matrix X:=[xij], i,j=1,2,...,n of interindustry transactions indicates the amount (x) produced by industry (i) and purchased by the industry (j) being an output for industry (i) and simultaneously an input for industry (j). The amount (xii) is produced and used within industry (i). The output xi could then be an intermediate or final

- Cella Guido (1988) “The Supply Side Approaches to Input-Output Analysis: An Assessment.” Ricerche Economiche, XLII, (3), pp. 433-451. - Deman, S. (1988) “Stability of Supply coefficients and consistency of supply-driven and demand-driven input-output models.” Environment and

Planning A, 20, pp. 811-816. - Bon, Ranko (1988) “Supply-Side Multiregional Input-Output Models.” Journal of Regional Science, 28 (1), pp. 41-50. - Penson, John B. Jr. & Hovav Talpaz (1988) “Endogenization of final demand and primary input supply in input-output analysis.” Applied

Economics, 20, pp. 739-752. - Oosterhaven, J. (1988) “On the plausibility of the supply-driven input-output model.” Journal of Regional Science, 28, pp. 203-217. - Adamou, N. (1988) Structural Analysis and Analysis of Structural Change in an Extended Input-Output Framework, Ph.D. Dissertation,

Department of Managerial Economics, Rensselaer Polytechnic Institute. - Dietzenbacher E. (1989) “On the relationship between the supply-driven and the demand driven input-output model.” Environment and Planning

A, Vol. 21 (11) pp. 1133-1539. - Gruver G. (1989) “On the plausibility of the supply-driven input-output model: a theoretical basis for input-coefficient change” Journal of Regional

Science Vol. 29, pp. 441-450. - Miller, R. E. (1989) “Stability of Supply coefficients and consistency of supply-driven and demand-driven input-output models: a comment.”

Environment and Planning A, 21, pp. 1113-1120. - Gruver Gene W (1989) “On the Plausibility of the Supply-Driven Input-Output Model: A Theoretical Basis for Input-Output Coefficient Change”

Journal of Regional Science, 29(3) pp. 441-450. - Rose, A. and T. Allison (1989) “On the plausibility of supply-driven input-output model: Empirical evidence on joint stability.” Journal of

Regional Science 29, pp. 451-458. - Oosterhaven J. (1989) “The Supply-Driven Input-Output Model: A New Interpretation but Still Implausible” Journal of Regional Science, 29(3)

pp. 459-465. - Beaumont P. M. (1990) “Supply and Demand Interaction in Integrated Econometric and Input-Output Models” International Regional Science

Review, 13(1&2) pp. 167-181. - McGilvray, James (1989) “Supply-Driven Input-Output Models” 9th International I-O Conference, Keszthely, Hungary. - Lorenzen, G. (1989) “Input-Output multipliers when data are incomplete or unreliable” Environment and Planning A, Vol. 21, pp. 1075-1092. - Adamou N. and J. M. Gowdy (1990) “Inner, final, and feedback structures in an extended input-output system.” Environment and Planning A, Vol.

22, pp. 1621-1636. - Dewhurst J. H. L. (1990) “Intensive income in demo-economic input-output models” Environment and Planning A, Vol. 22, pp. 119-128. - Miller R. E. & G Shao (1990) “Spatial & sectoral aggregation in the commodity-industry multiregional input-output model” Environment and

Planning A, Vol. 22, pp. 1637-1656. - Chen, C. Y. and A. Rose (1991) “The Absolute and relative Joint Stability of Input-Output Production and Allocation Coefficients.” in W.

Peterson (eds.) Advances in Input-Output Analysis: Technology, Planning, & Development, Oxford Univ. Press, pp. 25-36. - Deman S. (1991) “Stability of Supply coefficients and consistency of supply-driven and demand driven input-output models: a reply” Environment

and Planning A, Vol. 23, pp. 1811-1817 - Gowdy J. (1991) “Structural Change in the USA and Japan: an extended input-output analysis.” Economic System Research, 3, pp. 413-423. - Lekakis J. N. (1991) “Employment effects of environmental policies in Greece” Environment and Planning A, Vol. 23, pp. 1627-1637. - Campisi D., A Natasi & A. La Bella (1992) “Balanced growth and stability of the Leontief dynamic Model: an analysis of an Italian Economy”

Environment and Planning A, Vol. 24, pp. 591-600. - Lieu S. & G. I. Treyz (1992) “Estimating the economic and demographic effects of an air quality management plan. The case of Southern

California” Environment and Planning A, Vol. 24, pp. 1799-1811. - Szyrmer J. M. (1992) “Input-output coefficients and multipliers from a total flow perspective” Environment & Planning A Vol. 24, pp. 921-937. 12 Miller R. E. and P. D. Blair (1985) Input-output Analysis: Foundations and Extensions, Prentice Hall, p.319.


product.13 Final products are demanded by private consumers (yC), government (y

G), business for

investment processes (yI), and foreign consumers (y

X) (exports). These items constitute the demand of

domestically produced commodities. Total demand (net output) also includes the demand for foreign commodities (y

M) (imports).14 Primary inputs are not produced within the interindustry structure, such as

that of labor. Primary inputs may be decomposed as salaries and wages indicated by (wj), indirect taxes by (tj), and other value added by (vj). The most detailed decomposition of final demand and primary inputs is desirable. Interindustry flows can be represented in physical or in value flows, both ways being equivalent with different problems associated with each one of them.15 What is required for a balanced table is that the value of total production is equal to the value of the total demand. The column stocks in the final demand balances the value of total demand with the value of total production. Final demand is decomposed into domestic demand of both domestic and foreign products Y

D=[ y

C + yG + y

I - yM ] and yX

foreign demand of domestic products.

The concepts of interindustry accounting are presented in Table 1.1, their mathematical formulation in Table 1.3, and a numerical example16 in Table 1.2. The value of industrial demand is the summation of the row elements of the transaction matrix xio and the cost of the intermediate inputs of an industry is x

ojT.

The cost of the intermediate inputs is not equal to the value of the intermediate demand, i.e. xio ≠ xoj

T. The summation of all elements of the transaction matrix x

oo indicates the total value of intermediate input being

equal to the total value of intermediate output, the equilibrium of the aggregate intermediate demand to the aggregate intermediate supply. The Yi,j=1,...,r := [ y

C yG yI yS yX yM ] final demand (net output) is where the sum of row elements yields to total sectoral final demand yi. and the summation of the column

elements y.r := iTY the aggregate decomposition of the GNP as C+G+I+X+M. The total value of aggregate demand can be decomposed in terms of industrial (rows) and origin (columns). The primary input matrix Hi=1,..s, j provides the value of primary input per industry h.j

T and hs. the value of the decomposed output. The sum of all elements is the total value of the supply factors h.. . A column vector i

indicates a vector of 1's as i = [1, 1, ... , 1]T.

Theoretically, we may define another area of the interindustry flows representing the connection between final demand and primary input outside of the interindustry purchases.17 I do not use that alternative for the sake of simplicity, although, I believe it is preferable because it is complete conceptually permitting

13 It is clearly that interindustry analysis is a production oriented approach as Christ presented, and not just a technical distinction between

endogenous and exogenous sectors, as Sigel proposed, and Copeland M. A. in his comparative comment emphasized. See NBER (1955) Input-Output Analysis, An appraisal. Studies in Income and Wealth., Vol. 18, Princeton University Press, p. 286. The term endogenous sector is equivalent to processing sector, and the term exogenous sector is equivalent to final demand or primary input sector.

14 Imports may be used as inputs in the production process, and then their proper position is not in the final demand since are not final products but intermediate products. If they are not able to be produced domestically but required in the production process they should be placed below the processing sectors as a row of the primary input. A model with positive import row in primary input matrix or a negative column in final demand matrix provides the same algebraic results, but certainly, the interpretation is different. An import matrix that could allocate imports to their proper destination could provide an explicit information of the affect of imports in the economic structure. Although this would be my preference, I keep imports as part of the final demand following the conventional data.

15 Yan (1969) Introduction to Input-Output Economics, Chapter 1, Holt and Winston, New York for the equivalence between the physical and value approach.

16 As a numerical example I use the one provided by Miller & Blair (1985) pp. 15-18 and 321. The only difference from the above example is that here primary input and final demand are decomposed matrices instead of an aggregate row and column vector as in Miller and Blair.

17 Such date are published by the Planning Bureau of The Netherlands Gecumuleerde Productiestruc-tururmatrices (GPS-matrices) 1969-1985 Centraal Planbureau, Interne Notice, 17 Februari 1989.


general examination of equilibrium and stability issues. The summation of column elements of matrix [X H]T provides the cost value of the gross sectoral production x.j

T, which is equal to the value of total demand xi., the summation of row elements of the matrix [X Y]. The sectoral equilibrium of the gross

output is x.jT= xi., the total value of aggregate gross demand equals the value of aggregate gross supply.

From this descriptive representation one can extract some elementary, but important structural information.18 One may compare intermediate input (output) to primary input (final demand), and sectoral production (total demand), at the sectoral and aggregate level.

Equilibrium conditions are indicated at the aggregate (scalar) level, or at the sectoral (vector) level. The disaggregate (industrial) equilibrium is given by the sectoral equality of gross production and of total demand.19

x.jT = xi. = x

Equilibrium was generated because of the additional stock column in the final demand matrix. Without the presence of this column the system would be in disequilibrium. Although there is equilibrium in the industrial gross output, which is equal to the value of total demand, there is disequilibrium at the industrial intermediate demand and intermediate requirement vectors (xio ≠ xio

T), as well as in the industrial primary

input final demand (yi. ≠ h.jT). The interindustry model transformed into disequilibrium disturbs the

accounting identities.

Aggregate (scalar) equilibrium conditions are given at three points. The intersectoral equilibrium (xoo

) indicates the value of total industrial intermediate demand equal to the total value of industrial intermediate requirements.

The gross output equilibrium indicates the equality between total demand and the gross value of production.

x.. = xoo

+ y.. = xoo

+ h.. Finally, from the above one can conclude that the net demand-net supply equilibrium

y.. = h.. known in national accounting as the value of final aggregate demand, is equal to the value of income (C + G + I + X - M = W + V + T). This equilibrium can be composed from an industrial level as

y.. = h.. ⇐ iTyi. = h.jTi

or from its macroeconomic representation as y.. = h.. ⇐ y.r

Ti = iThs. Special frames in Tables 1.2 and 1.3 indicate the points of equilibrium.

18 Santhanam K. & R. Patil (1972) “A study of the Production of the Indian Economy: An International Comparison” Econometrica, 40 (1),

January, pp. 159-176. 19 In this work we assume general equilibrium from this point and thereafter gross output will be indicated without any subscript.

xoo:= iΤXi


Table 1 Interindustry Accounting

Interindustry Transaction Matrix

nxn Intermediate

Industrial Demand col. vector

Final Demand Matrix nxr

Total Industrial Final Demand col. vector

Total Industrial Demand

(Equilibrium Vector) col. vector

Intermediate

Industrial

Requirements row

vector

Total Value of Intermediate

Industrial Requirements

& simultaneously of Intermediate

Industrial Demand (Equilibrium Point)

scalar

Final Demand

Decomposed by type

row vector

Total Value of Final

Demand

(simultaneously of the

above col. and the left

row))

scalar

Total value of Gross

Production

(Equilibrium Point)

scalar

Primary Input Matrix rxn

Total Primary Input by type col. vector

Total Primary Input by industry

row vector Total value of Primary Input

(simultaneously of the above col. and the left

row)) scalar

Total Industrial Value of Production

(Intermediate Requirements & Primary Inputs)

(Equilibrium Vector) row vector

Total Value of Production

(Equilibrium Point) scalar


Table 1.2 Interindustry Accounting

Xij xio

Yir yi.

xi. xi. := xio + yi.

xoj

T

xoo

y.rT

y.r := iTY C, G, I, X, M

y..

y.. := iTYi

x.. x.. := iTxi.

x.. := x.jTi.

Hsj

hs.

hs. := Hi W, T, V

h.jT

h.jT :=iTH

h..

h.. := iTHi

x.jT

x..

The matrix

provides the data framework of interindustry accounting. Social Accounting Matrices20 (SAM), and the System of Economic & Social Accounting Matrices & Extensions21 (SESAME) provide an extension of 20 Pyatt, G. (1991) “Foundamentals of social accounting” Economic System Research 3, pp. 315-341.

x io:= Xijj=1

n

∑xio:= Xi

y i.:= Yijj=1

n

∑ ,∀ι

y i.:=Yi

x ojT

:= Xiji=1

n

∑xoj

T

= iΤX

xoo:=i=1

n

∑ Xijj=1

n

∑

xoo:= iΤXi

y .r:= Yirr∑ ,∀r y.. := Yir

r∑

i∑

x..:= xoo + y..

hs. := Hsjj=1

n

∑ ∀r

h. j

T

:= Hsjs∑ ∀j h..:= Hsj

j∑

s∑

x. jT= xoj

T

+ h. j

T x.. = xoo + h..

X Xi Y Yi Xi +YiiTX iTXi iTY iTYi iTXi + iTYiH Hi 0 0 0iTH iTHi 0 0 0

iTX + iTH iTXi + iTHi 0 0 0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥


the above-described scheme integrating national income and product accounts with interindustry accounts. A numerical example matrix is indicated in Table 1.3.

Table 1.3 Interindustry Accounting Matrix

Transactions Ind. 1 Ind. 2 T. Int. D. F. D. 1 F. D. 2 T. F. D T. D. Ind. 1 150 500 650 50 300 350 1000 Ind. 2 200 100 300 1200 500 1700 2000

T. Int.Req. 350 600 950 1250 800 2050 3000 Prim. In. 1 150 120 270 Prim. In. 2 500 1280 1780 T. Pm. In. 650 1400 2050 T. V. P. 1000 2000 3000

3 Descriptive Coefficients in Interindustry Modelling

We have three data matrices: the interindustry transactions X, the final demand Y, and the primary inputs

H, as components in a data matrix from where we derive four22 different types of descriptive

coefficients in an interindustry system of accounts.

The first type of descriptive coefficients provides the distributions of each column of the data matrix. The part [X, H]T is divided by the sectoral gross output x, while Y by the row vector iTY indicating the respective summation of the column elements. Part of the first distribution is the traditional input coefficient matrix A. The second type of descriptive coefficients views the same data row-wise. The allocation coefficients matrix B is part of the output distribution matrix. Obviously, B indicates different descriptive information than that of A, but one cannot accept linearity in allocation with more difficulty than linearity in production. Matrices A and B indicate the linearity analytical assumption in production and allocation interindustry models. Matrices Ya and Hb provide one kind of exogenous determination in

the demand and supply interindustry approaches. Matrices Ya and Hb standardize all types of final demand and primary input while Yc and Hc distribute a unit of final demand and primary input. Therefore matrices Yc and Hc may also play a role as exogenous elements in interindustry modeling with a different meaning

than that of Ya and Hb.

The two-dimensional distribution matrices Yc and Hc have the same average, which is also true with matrices Yd and Hd. Matrices Yc and Yd as well as matrices Hc and Hd have the same shape, therefore the same coefficients of Skewness and Kurtosis. 21 Keuning Steve J. (1994) “The SAM and Beyond: Open, SESAME!” Economic System Research, Vol. 6., No. 1, pp. 21-50. The same paper was

presented at the International Conference for Research in Income & Wealth in Canada, August 1994. 22 Any vector indicated inside the symbols < > implies a diagonal matrix defined based on an indicated vector. Superscript ‘a’ indicates “demand

driven analytical coefficients” or in more general way column distributions. Superscript ‘b’ indicates “supply driven analytical coefficients” or row distributions. Superscript ‘c’ indicates two-dimensional distributions of matrices X, Y, and H with respect to the overall total value of each one. Superscript ‘d’ indicates two-dimensional distribution with respect to the scalar value of total gross output.

X YH 0⎡

⎣ ⎢ ⎤

⎦ ⎥


Matrices [A, Ha]T and [D, Hd]T as well as [B, Yb] and [D, Yd] give the distribution of gross output in a different perspective. The one-dimensional distributions provide the sectoral gross output (vector) distributed in each sector, and the two-dimensional distributions the total gross output (scalar).

Table 2 Typology of Descriptive Coefficients

Description Data Reference

Descriptive Coefficient

Matrices

Column

Distributions

Matrices

1. A unit of sectoral gross output to sectoral requirements

2. A unit of each Final Demand type

x

iTY

Row

Distributions

Matrices

1. A unit of sectoral total demand to sectoral allocations

2. A sectoral allocation of a unit in each type of primary inputs

x

Hi

Two-dimensional Matrices of

Interm.Trans. Prim. Inputs & Final Demand

1. % of interindustry transaction with respect to total intermediate transactions

2. % of primary inputs with respect to total value added

3. % of final demand with respect to total final demand

iTXi

iTHi

iTYi

Two-dimensional Gross Output

Matrices

1. % of demand with respect to total demand 2. % of value of production with respect to total

value of production

iTx

iTx

XH⎡

⎣ ⎢ ⎤

⎦ ⎥

Y[ ]

AHa⎡

⎣ ⎢ ⎤

⎦ ⎥

Ya[ ]

X Y[ ]H[ ]

B Yb[ ]Hb[ ]

X[ ]H[ ]Y[ ]

C[ ]

Hc[ ]Yc[ ]

X Y[ ]XH⎡

⎣ ⎢ ⎤

⎦ ⎥

D Yd[ ]DHd⎡

⎣ ⎢ ⎤

⎦ ⎥


3.1 Column distributions matrices

Column distributions23 show the requirements of total sectoral production unit and the sectoral

distributions in final demand [Ya]. The traditional input coefficient matrix A indicates linearity in the production model. Explicit knowledge of A and x implies knowledge of primary inputs.

The matrix Ha reveals the share of primary input to total production. The summation of column elements of the input coefficients and the share of primary input to total value of production equals to one.24

The summation of the column elements of matrix Ya yields one.25

Descriptive coefficients of the traditional demand driven model are derived whenever we post multiply the interindustry accounting matrix by the inverse diagonal of the value of total production and final demand.

In a more general form we have the following matrix:26

23

24 iT[A, Ha ] = iTX<x>-1 + iT H<x>-1 = (iT X + iT H)<x>-1 =

= (xT I)<x>-1 = <xT ><x>-1 = I

25 iTYa = iTY<iTy>-1= I 26 The second & fourth column of the general matrix is NOT the summation of the rows of the first & third columns. Their numerical results are

calculated from the original data.

AHa⎡

⎣ ⎢ ⎤

⎦ ⎥

Aiji∑ + Hsj

a

s∑ = 1∀j

Yira

i∑ = 1∀r

X YH 0⎡

⎣ ⎢ ⎤

⎦ ⎥ < x >−1 00 < iTy> −1

⎡

⎣ ⎢ ⎤

⎦ ⎥ =A Y a

Ha 0⎡

⎣ ⎢ ⎤

⎦ ⎥

A =

X11

x1LX1n

xnM O MXn1

x1L

Xnn

xn

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Ha =H < x > −1=

T1x1

L Tnxn

W1

x1L Wn

xnV1x1

L Vn

xn

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

Y a = Y < iTy >−1=

Yc1YcMYcnYc

Yg1Yg

L Ym1

YmM O MYgnYg

L YmnYm

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥


.

In numerical form this matrix displays the column distributions in Table 3. The demand driven-production model is based on these coefficients. Here we have distributions of a value unit in each one of the industries with respect to their intermediate and primary input requirements. The following figures indicate

graphically these distributions. Figure 1 indicates the data from matrix27 , and Figure 2

data from matrix . Figure 1

Table 3

Column Distributions

Input Coefficients Ind. 1 Ind. 2 T. Int. D. F. D. 1 F. D. 2 T. F. D T. D. Ind. 1 0.1500 0.2500 0.2167 0.0400 0.3750 0.1707 0.3333 Ind. 2 0.2000 0.0500 0.1000 0.9600 0.6250 0.8293 0.6667

T. Int. Req. 0.3500 0.3000 0.3167 1 1 1 1 Prim. In. 1 0.1500 0.0600 0.0900 Prim. In. 2 0.5000 0.6400 0.5933 T. Prm. In. 0.6500 0.7000 0.6833

T. V. P. 1 1 1

27 The second column of the matrix indicates within parentheses the matrix operation from where we derive the coefficients. Upper scripts indicate

the type of descriptive information. For the sake of simplicity, the matrix operation won't be indicated but only the type of descriptive coefficients and the data based on. It is obvious that in our graphical figures we do not include the second and fourth row containing totals of the respective matrices.

A (Xi < x >−1 )a Ya (Yi)a (Xi+ Yi)a

iTA (iTXi < x >−1)a 1 1 1Ha (Hi < x >−1)a 0 0 0

(iTH)a (iTHi < x >−1)a 0 0 01 1 0 0 0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

A (Xi<x >−1 )a

Ha (Hi<x >−1 )a

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

Ya (Yi)a (Xi+Yi )a[ ]

Input distribution of a gross production unit in allindustial sectors

Industrial sectors & Intermediate Demand

Inpu

t Disr

ibut

ion

00.20.40.60.8

1

Ind 1 Ind 2 Intr. Dem

PI 2

PI 1

Ind 2

Ind 1


Figure 1 illustrates the requirement distributions in each industry in terms of a produced value unit. All industries are standardized in terms of their production value.

Matrix Ya is significant for policy analysis indicating the sectoral distribution of a unit in each type of final

demand. We may simulate policy alternatives by changing the coefficients of the Ya matrix. Matrix Ya provides an alternative to the traditional output multiplier assumption. The traditional multiplier assumption is useful for analytical purposes by isolating or standardizing the ejection of each industry. The unitary multiplier, for example, yields the total production of a given type of demand, i.e. public consumption, distributed as a unit in sector one and zero in all other sectors. The uniform multiplier indicates the impact on total production due to a unit of demand in all sectors. These analytical advantages are impractical and misleading from a policy point of view. None of theses alternatives assimilates any realistic situation. Matrix Ya distributes a unit of a specific final demand to various industrial sectors. Let us take the public consumption as an example, assuming its sectoral distribution given by the second final demand column in our numerical example. A value unit of public consumption is distributed by 37.50% to the first and by 62.50% to the second industry. Changing this distribution is indeed a policy matter and objective. We are interested in evaluating the impact of the existent distribution, and the impact of any other alternative distributions we would like to consider. The impact distributions provided in matrix Ya is much more valuable than the impact of a unitary distribution (1 and 0's) or a uniform distribution (all 1's). We are not interested in demand distributed as a unit to one industrial sector and none to all others, neither is demand equal to all industrial sectors.

Figure 2

Matrix Ya evaluates the distributions of a unit in each decomposed aspect of the final demand. Matrix Ya used as a weight to the multiplier matrix provides one type of decomposed multipliers. Standardized final demand requires special care because it may also mislead. Private consumption is not at the same level as investment for example, and matrix Ya by treating i.e. private consumption and investment equally does not take this fact into account.

Distribution of a Final Demand

Types of Demand

Fina

l Dem

and

Dist

ribut

ion

0

0.2

0.4

0.6

0.8

1

FD 1 FD 2 FD TotDem

Ind 2

Ind 1


3.2 Row distributions matrices28

Row distributions matrices provide descriptive information of a sectoral total demand unit for industries by

matrix [B, Yb] and by matrix Hb the industrial distributions of a unit in each type of primary input. The analytical assumption of the allocation model (linearity in allocation) is given by matrix B. The matrix of the final demand share to total demand is given by Yb. The matrix of sectoral distributions in primary

inputs, given by Hb is useful for policy evaluations in an analogous way to matrix Ya, since its coefficients reflect the sectoral composition of a unit in a specific type of primary inputs.

The summation of the row elements in matrix [B, Yb] equals one for all i rows.29

The summation of the row elements in matrix Hb is one for all s rows.30

The same data matrix premultiplied by the diagonal inverse matrix of the row summations

provides the allocation coefficients.

Transactions are presented either as required input for all interdependent processing sectors and primary input rows, or as an allocated output between the processing sectors and final demand. Table 4, the respective matrix and Figures 3 & 4 give numerical, mathematical and graphical information of the allocation descriptive coefficients. The distribution of a unit of total demand to intermediate and final demand given in Figure 3 is the analytical assumption of the allocation model. It is obvious that the main

28

29 [B, Yb ]i = Bi + Ybi = <x>-1Xi +<x>-1Yi = <x>-1(Xi + Yi) = <x>-1I<x> =

= <x>-1<x> = I

30 Hbi = <Hi>-1Hi= I.

Bijj∑ + Yir

b

r∑ = 1∀i

Hsjb

j∑ = 1∀s

X YH 0⎡

⎣ ⎢ ⎤

⎦ ⎥

< x >−1 00 < hi >−1

⎡

⎣ ⎢ ⎤

⎦ ⎥ X YH 0⎡

⎣ ⎢ ⎤

⎦ ⎥ =B Yb

Hb 0⎡

⎣ ⎢ ⎤

⎦ ⎥

B =

X11

x1LX1n

x1M O MXn1

xnLXnn

xn

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Yb =< x >−1 Y =

Yc1x1MYcnxn

Yg1x1

L Ym1x1

M O MYgn

xnL Ymn

xn

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Hb =<Hi >−1 H =

T1T j

L TnT j

W1

Wj

L Wn

WjV1V j

L Vn

V j

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥


diagonal elements in both matrices A and B are the same. Deman shows that for every non-main diagonal

element of the matrices A and B the relationship exist.31 These two points yield to the fact

that matrices A and B have the same determinant.32 Therefore, although matrix B indicates different descriptive information than matrix A, both provide the similar information described from a different point of view. Figure 3 graphs the distributions of a sectoral total demand unit. The first industry supplies 50.00% of its output to the second one and only 35.00% is allocated to final demand, mainly to the second type of final demand by 30.00%. On the contrary, the pattern of the second industry is quite different. The second industry allocates 85.00% to final demand, distributed by 60.00% to the first and 25.00% to the second type of final demand.

Table 4 Row Distributions (Allocation coefficients)

Output

Coefficients Ind. 1 Ind. 2 T. Int. D. F. D. 1 F. D. 2 T. F. D T. D. Ind. 1 0.1500 0.5000 0.6500 0.0500 0.3000 0.3500 1 Ind. 2 0.1000 0.0500 0.1500 0.6000 0.2500 0.8500 1

T. Int. & D 0.1167 0.2000 0.3167 0.4167 0.2667 0.6833 1 Prim. In. 1 0.5556 0.4444 1 Prim. In. 2 0.2809 0.7191 1 T. Prm. In. 0.3171 0.6829 1

T. V. P. 0.3333 0.6667 1

Figure 4 depicts the industrial distributions of primary inputs, decomposed and total, as well as the total value of production. This Figure includes graphical information of matrix Hb, and rows (iTH)b and (iTX+iTH)b.

We need to be careful33 since matrices A, Ha and Ya read only column wise and B, Hb and Yb read row

wise. Matrices Hb and Ya denote policy objectives and matrices Ha and Yb register policy results.

31 Deman S. (1988, op. cit.) p. 813.

32 If 1) diag(A) = diag(B) and 2) ∀ non-diag i,j ∃ , then det(A)=det(B). Adamou N. (1988, op. cit.)

33 Cronin F. J. (1984, op. cit.) for example formulates a demand driven model with allocation function indicated as x=(I-<i'B>)-1

f (p. 524) in table 1 and formula 11. The term <i'B> indicates a summation of column elements in matrix B. Based on this allocation model he presents the multiplier

M6=(I-<i'B>)-1

i (formula 17, p. 525).{ In the printed text multiplier M6 as well as the Leontief multiplier M2 are indicated without the inverse notation but calculations are proper} This multiplier provides values greater than 10 for 19 US. industrial sectors, with a value of 454.714 for the sector 67, Radio & TV Broadcasting. In p. 528 Cronin states that “M6 employs derived information from each row of B separately” although his numerical results are based upon the mathematically incorrect calculations of formulas 11& 17 using the column instead of row summations of matrix B.

aij =xixjbij

B Xi( )b Yb Yi( )b 1iTX( )b iTXi( )b iTY( )b iTYi( )b 1Hb 1 0 0 0iTH( )b 1 0 0 0

iTX + iTH( )b 1 0 0 0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

aij =xixjbij


Matrices Ya and Hb provide the distributions causing production. Matrices, or specific columns and rows respectively of Ya and Hb can be used in policy simulations. Given sectoral distributions in public expenditures and taxes, one can evaluate the sectoral gross output multiplier of a balanced budget.34 Having the sectoral distribution in investment and wages one can explore their feedback.35 One can examine the different impact of imports and exports on gross output.36

Figure 3

The negative column of imports effects the row summation of the final demand matrix and the sectoral value of gross output. It is not unusual to observe negative total final demand due to larger demand for imports than the domestic demand of domestic products in a given industry. A negative element of total final demand indicates a local stability problem.37 If imports are greater than domestic production then we

- Oosterhaven (1988, op. cit.,) in his footnote 3 (p. 208) presents correctly the demand-constrained Ghoshian model as x=(I-<Bi>)

-1v, and

criticizing the nature (diagonal matrix does not provide interrelations) of the demand constrained Ghoshian model properly concludes that this models as well as the supply-constrained Leontief model are not too useful.

- Adamou N. (1991) “Clarifying Analytical Assumptions in Inter-industry Modelling” Workshop in interindustry economics, Sage Graduate School, Troy, NY, August.

34 Adamou N. (1992) “Inter-Industry Structure in New York State and its implication for fiscal policy, with a particular emphasis on taxes.” New York State Tax Study Legislative Commission, April.

- Adamou, N. (1991) “Fiscal Policy , Foreign Trade and Industrial Structure in Japan: 1960-1985.” Ways & Means Committee of the New York State Assembly, December.

- Adamou N. (1989) “Interindustry Analysis and Budgeting Process .” Ways & Means Committee of the New York State Assembly, November. 35 Adamou N. (1990) “Capital and Labor Feedback Structures in an Extended Input-Output System: A Comparison of the United States and Japan.”

International Symposium on Economic Modelling, University of Urbino, Italy, July 23-25. 36 Adamou, N. (1991) “Industrial Impact of Imports and Exports in Japan: 1960-1985.” International Conference of the International Trade &

Finance Association, May 31-June 2, Marseilles, France. 37 Stability in terms of aij and bij requires the coefficients to be less than one and positive. This implies that the vectors of total final demand and

total primary input are positive or non negative. - Takayama, A. (1987) Mathematical Economics, 2nd ed., Cambridge U. Press, p. 360 ( - the existence problem ) assumes positive final demand. If

there is a negative element in the total final demand vector due to a great value of imports, then the respective sum of bij row is greater than one. Cronin (1984, op. cit., p. 524, footnote 3) comments that the demand driven output multiplier is negative ( ∂xi/∂fi<0 ). Cronin also assesses that in the case the industrial output is allocated completely into processing sectors and not at all to the final demand, i.e. the sum of bij coefficient of the respective row is one, then the demand driven output multiplier( ∂xi/∂fi ) is undefined. We may also face the opposite case, an industrial sector allocating all its output to final demand (shoes). In this case the sum of the respective interindustry allocation coefficient row is zero, and the matrix does not have full rank. Very rarely a negative element may appear in a transaction table as those reported for the 1960, 1965 & 1970 Japanese I-O tables. See:

- Uno, K. (1989) Measurement of Services in an Input-Output Framework, Amsterdam, North-Holland. Such cases were examined by

Distribution of a unit of Total Demand

Industries

Dis

tribu

tion

0

0.20.4

0.60.8

1

Ind 1 Ind 2 T_Dem

FD 2

FD 1

Ind 2

Ind 1


have further problems indicated by a negative element in the gross output column, which is a local instability problem. These problems, however may be more easily detected in row distributions, due to the balancing aspect of interindustry accounts that have impact on the respective columns. The importance of the stability question requires detailed discussion, which will be presented later.38

Figure 4

The interrelation of final demand and primary inputs (income aspect in national accounts) provides different dimensions to the stability problem requiring a mathematical investigation and interpretation to each one of the respective cases. For example, we need to examine under which conditions a negative sectoral total final demand due to increased value of imports may cause or not a subsidy or loss (negative element in primary input).

- Adamou N. (1991) “Negative Elements in Japanese Input-Output Tables and the Problem of Stability.” International Symposium of Economic

Modelling, University of London, July 9-11. - Adamou N. (1990) “The Impact of Negative Total Sectoral Final Demand on the Production and Allocation Structure in Interindustry models.”

Workshop in interindustry analysis, Sage Graduate School, November. 38 Examples with various stability problems and their respective descriptive, structural and distribution matrices are provided in appendixes

Industry Distributions of a Unit in Primary Inputs &Total Production

Types of Primary Input and TotalProduction

Dis

tribu

tion

0

0.5

1

PI 1 PI 2 PI TotProd

Ind 2

Ind 1


3.3 Two-dimension distribution matrices of Intermediate Transactions, Final Demand and Primary Inputs

This class of two-dimension distribution matrices39 indicates the relationship of each element in matrices X, Y, and H to the respective overall total in each matrix. An element cij in matrix C reveals the percentage of industry (j) purchases from industry (i) with respect to total intermediate interindustry transactions. The element yij in matrix Yc provides the percent that industry (i) sales to final demand type

(j) with respect to all industrial sales to all types of final demand. The element hij in matrix Hc gives the percent that industry (j) requires from the primary input (j) relative to all industrial requirements from all types of primary inputs. As an example, matrix Yc furnishes the percentage of value in metallurgical sector's private investment related to total value in final demand, while matrix Hc marks the percentage metallurgical sector's wage contribution with respect to the total value of primary inputs.

Table 5 displays the numerical example of this class two dimension distribution matrices. The distribution matrices C, Hc and Yc have the same denominator throughout in each matrix permitting numerical operations row wise as well as column wise simultaneously. The marginal distributions of the interindustry transaction matrix indicate a unit value in intermediate transaction distributed either as intermediate industrial demand or as intermediate industrial requirements. Marginal distributions in final demand and primary input matrices hold a similar meaning. The column marginal distribution of final demand in Table 5 appears also in Table 3, but Table 3 does not contain the marginal distributions of intermediate transactions and primary inputs. By the same token, the row marginal distribution of primary inputs in Table 5 appears also in Table 4.

The importance of this particular class of two-dimension descriptive information is that we move on the standardization from columns or rows to the matrix itself. This allows one to compare among any element throughout each one of the matrices, C, Yc and Hc.

One can scrutinize for example the distribution of final demand unit into 60.98% in consumption (2.44% in the first industrial sector & 58.54% in the second industrial sector) and 39.02% of investment (14.63% in the first industrial sector & 24.39% in the second industrial sector), or otherwise as 17.07% demand from the first industrial sector and 82.93% from the second industrial sector.

39

C =

X11

Xij

j=1

n

∑i=1

n

∑L

X1n

Xij

j=1

n

∑i=1

n

∑M O MXn1

Xij

j=1

n

∑i=1

n

∑L

Xnn

Xij

j=1

n

∑i=1

n

∑

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

(Y) c =

Y 11

Yij

j=1

r

∑i=1

n

∑L

Y 1r

Yij

j=1

r

∑i=1

n

∑M O MYn1

Yij

j=1

r

∑i=1

n

∑L

Ynr

Yij

j=1

r

∑i=1

n

∑

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

(H)c =

H11

Hij

j=1

n

∑i=1

s

∑L

H1n

Hij

j=1

n

∑i=1

s

∑M O MHs1

Hij

j=1

n

∑i=1

s

∑L

Hsn

Hij

j=1

n

∑i=1

s

∑

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥


Similarly, one can categorize the distribution of a unit in primary inputs as 13.17% paid wages (7.32% in the first industrial sector & 5.85% in the second industrial sector) and 86.98% as other value added (24.39% in the first industrial sector & 62.44% in the second industrial sector), that is 31.71% of primary input unit are allocated to the first industrial sector and 68.29% to the second industrial sector.

Table 5 Two-Dimensional Distributions of Intermediate Transactions, Final Demand and Primary Inputs

Ind. 1 Ind. 2 T. F. D. 1 F. D. 2 T. F. D

Ind. 1 0.1579 0.5263 0.6842 0.0244 0.1463 0.1707 Ind. 2 0.2105 0.1053 0.3158 0.5854 0.2439 0.8293

T. 0.3684 0.6316 1 0.6098 0.3902 1 Prim. In. 1 0.0732 0.0585 0.1317 Prim. In. 2 0.2439 0.6244 0.8683

T. 0.3171 0.6829 1

The Table 5 given as a matrix

provides three two-dimensional distribution matrices and the resulting marginal distributions. The matrix

composition of the two-dimensional matrices is . Each one of these matrices is

pictured in Figures 5, 6, and 7.

Figure 5

Figure 5 exhibits the fact that among all intermediate industrial transactions the 52.63% are the purchases of the second industrial sector from the fist one. Figure 6 displays the two-dimensional distribution of a primary input unit. This distribution indicates the relative dominance among all primary inputs the other value added of the second industrial sector, been the 62.44% of the total primary inputs. Figure 7 exposes

C Xi( )c Y( )c Yi( )c

iTX( )c 1 iTY( )c 1H( )c Hi( )c 0 0iTH( )c 1 0 0

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

X<i TXi> −1 Y<i TYi> −1

H<i THi>−1 0

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

Ind 1Ind 2

PurchasesIn

d 1

Ind

2Sales

0

1

Two Dimensional Distribution of an Interindustry Transactions Unit

The Inte rindustrytransaction wi th the

most influence


the strength of private consumption demanding output from the first industrial sector (58.54%) relatively to weak private consumption in the second industrial sector, while public consumption is 1.6 times more in the second industrial sector than the first one.

The distribution matrices Yc and Hc are important at the multiplier decomposition process. The matrix Yc indicates the decomposition of a final demand unit to various final demand parts in their industrial decomposition. This matrix is more realistic for the multiplier matrix decomposition than Yb since each type of the final demand is not treated equally. Policy initiatives on the demand side may be seen as alterations of the Yc matrix. Matrix Hc serves as a supply side (income) unit decomposition.

Both Yc and Hc matrices provoke change in gross output, indicating either existent decomposition of demand and income or respective policy targets.

Figure 6

Figure 7

Ind1

Ind2

PI 1PI 20

0.5

1

Two Dimensional Dis tribution of a Primary Input Unit

Relative Dominancein Primary Inputs

FD1

FD2

Private &Public

Consumption

Ind 1Ind 20

0.5

1

Two Dimensional Distribution of a Final Demand Unit

Private Consumpti on of thesecond industria l sector is

58.54%

Industrial sect or 2absorbs 1.6 timesmore of the publicconsumption than

the first one .


3.4 Two-dimensional Total Gross Output Distribution Matrices

Two-dimensional total gross output distribution matrices measure all elements of the interindustry accounting as a percentage of the total gross output. Two two-dimensional distribution matrices view the gross output unit as requirement and as demand. Interindustry transactions are a common area to both. Table 6 provides this information.

Table 6 Two-Dimensional Distribution of the Gross Output

Ind. 1 Ind. 2 T. F. D. 1 F. D. 2 T. F. D T. D.

Ind. 1 0.0500 0.1667 0.2167 0.0167 0.1000 0.1167 0.3333 Ind. 2 0.0667 0.0333 0.1000 0.4000 0.1667 0.5667 0.6667

T. Int. Req. 0.1167 0.2000 0.3167 0.4167 0.2667 0.6833 1 Prim. In. 1 0.0500 0.0400 0.0900 Prim. In. 2 0.1667 0.4267 0.5933 T. Prm. In. 0.2167 0.4667 0.6833

T. V. P. 0.3333 0.6667 1

This indicates40 the distribution matrix of the demand side of the gross output [D Yd] and the requirement side of the gross output matrix [D Hd]T. From interpretation point of view matrices D Yd and Hd relate the

40

D =

X11

Xij +j=1

n

∑ Yij

j=1

r

∑i=1

n

∑i=1

n

∑L

X1n

Xij +j=1

n

∑ Yij

j=1

r

∑i=1

n

∑i=1

n

∑M O MXn1

Xij +j=1

n

∑ Yij

j=1

r

∑i=1

n

∑i=1

n

∑L

Xnn

Xij +j=1

n

∑ Yij

j=1

r

∑i=1

n

∑i=1

n

∑

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

(Y)d =

Y11

Xij +j=1

n

∑ Yij

j=1

r

∑i=1

n

∑i=1

n

∑L

Y 1r

Xij +j=1

n

∑ Yij

j=1

r

∑i=1

n

∑i=1

n

∑M O MYn1

Xij +j=1

n

∑ Yij

j=1

r

∑i=1

n

∑i=1

n

∑L

Ynr

Xij +j=1

n

∑ Yij

j=1

r

∑i=1

n

∑i=1

n

∑

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

(H)c =

H 11

Xij +j=1

n

∑i=1

n

∑ Hij

j=1

n

∑i=1

s

∑L

H 1n

Xij +j=1

n

∑i=1

n

∑ Hij

j=1

n

∑i=1

s

∑M O MHs 1

Xij +j=1

n

∑i=1

n

∑ Hij

j=1

n

∑i=1

s

∑L

Hsn

Xij +j=1

n

∑i=1

n

∑ Hij

j=1

n

∑i=1

s

∑

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥


interindustry accounting data to the value of gross output, while the matrices C Yc and Hc present the relationship with respect to their own summation. As a result one may view matrices Yc and Hc as policy indicators and Yd and Hd as policy results and matrices Yc and Hc as policy indicators. A distribution of a final demand unit (Yc) has impact in primary inputs' relationship to gross output (Hd), and vice versa, a distribution of a primary input unit (Hc) has impact in final demand's relationship to gross output (Yd).

The above numerical data are given in a matrix form as

.

Each one of the two dimensional distributions are pictured in the followed Figures 8 and 9. It is important one to observe that the shape of these distributions is the same as those pictured in Figures 5, 6 and 7.

Figure 8

D Xi( )d Yd Yi( )d Xi+ Yi( )d

iTX( )d iTXi( )d iTY( )d iTYi( )d 1Hd Hi( )d 0 0 0iTH( )d iTHi( )d 0 0 0

iTX + iTH( )d 1 0 0 0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

Ind

1In

d 2

Ind

1In

d 2

0

0.2

Two dimensional distribution of a Gross Output Unit in InterindustryTransactions

This 31.67% of gross outputdistribution to interindustrytransactions has the sameshape as the distribution of aunit in interindustrytransactions. (picture 5)

Marginal Distributions Sales Purchases 1) 68% or 22% 37% or12% 2) 32% or 10% 63% or 20%


Figure 9

Figure 10

Table 7 Descriptive Statistics in two-Dimensional Distributions

Yc Yd Hc Hd Avg. 0.2500 0.1708 0.2500 0.1708 Std. 0.2409 0.1646 0.2634 0.1800 Skew 1.1974 1.1974 1.4567 1.4567 Kurt 1.6892 1.6892 1.7141 1.7141

The central location measured by the average is the same in matrices Yc and Hc as well as in Yd and Hd. This is an indication of their equilibrium between final demand and primary inputs besides their different desegregation. The correlation coefficient of the final demand columns and primary inputs rows provide evidence of opposite direction in their associationship. This may be viewed as an indication that final demand and primary inputs provide the same information from a different point of view. Skewness and kurtosis as measurements of shape are the same in the two dimensional distributions.

Ind1

Ind2

PI 1PI 2

0

0.5

Two dimensional distribution of a Gross Output Unit in Primary Inputs

This 68.33% of gross outputdistribution to primary inputsis the same as the distributionof a primary inputs unit.(picture 6)

FD1

FD2

Ind 1Ind 20

0.2

0.4

Two Dimensional Distribution of a Gross Output unit in Final Demand

This 68.33% distribution ofgross output to f inal demand

has the same shape as thedistribution of a final demand

unit. (picture 7)


4 Interindustry Demand & Supply Models and Equivalence transformations

4.1 Same Solution from ‘Demand’ and ‘Supply’ Models

Given demand and production structures the Leontief model provides gross output. The matrix of interindustry transactions X, the vector of final demand y, and a linearity assumption are utilized.41 Linearity is a harsh assumption not easily acceptable even in production, and criticized by Leontief.42

The Leontief system is represented by a) the accounting identity of a sectoral distribution in total demand; b) the analytical assumption of linearity in production, i.e. the value of total demand of an industry is proportional to its production requirements; and c) the model specified by the Leontief inverse matrix. The model means that for a given level and composition of final demand, and with the given technology, we can determine the appropriate level and mix of industrial production. This determination is evaluated by the Leontief inverse, providing explicitly the interrelations among industrial sectors.

x ≡ Xi + Yi A = X<x>-1

x = [I-A]-1Yi = ZYi

The same value of industrial production is determined also by the Ghoshian system.43 Accounting identities present equalities between value of total industrial production and value of total demand in each industrial sector. The accounting identity in a final demand driven44 Leontief model provides the allocation of demanded gross output. The linearity assumption in the input requirements of the same model is linked to the allocation of the total demand. A different point of view presents the primary inputs driven model. The accounting identity in the Ghoshian model presents the production requirements linked to the linearity assumption of interindustry output allocation proportional to gross output. This reveals a clear and obvious symmetry between the two systems. As the Leontief model relates the allocation of final demand

41 Samuelson, Paul, A. (1952) “A Theorem Concerning Substitutability in Open Leontief Models” and

Georgescu-Roegen, Nicholas (1952) “Some Properties of a Generalized Leontief Model” , both in Activity Analysis of Production and Allocation. 42 Leontief W. (1943) “Exports, Imports, Domestic Output and Employment”, The Quarterly Journal of Economics, February.

Leontief W., (1946) “Wages, Profits and Prices”, The Quarterly Journal of Economics, November. 43 Solutions to Leontief and Ghoshian models are provided in matrix form. The solution of the demand driven model is:

x ≡ Xi + Yi ⇒ x = A<x>i + Yi ⇒ x = Ax + Yi ⇒ Ix - Ax = Yi ⇒

⇒ [I-A]x = Yi ⇒ x = [I-A] -1Yi The solution of the supply driven models is:

xT ≡ iTX + iTH ⇒ x ≡ XTi + HTi

xT = iT<x>B + iT H ⇒ x = (<x>B)T i + HT i = BT<x>i + HT i xT = xTB + iTH ⇒ x = BTx + HTi xTI− xTB = iTH ⇒ Ix − BTx = HTi xT [I-B] = iTH ⇒ [I-BT] x= HTi xT = iTH [I-B] -1 ⇒ x=[I-BT] -1 HTi The two linear systems of equations provide the same solution.

44 The exogenous driving elements in the Leontief model is the vector or matrix of final demand and in the Ghoshian model the vector or matrix of primary inputs. It is not a demand or a supply function as an interrelationship of quantities to prices. None of the models assumes anything about price elasticities. For this reason the names final demand and primary inputs driven systems will be used instead of the traditionally accepted demand and supply driven systems.


to production requirements the Ghoshian model relates primary input requirements to interindustry (intermediate) output allocation.

In the Leontief system the accounting identity decomposes the total demand while in the Ghoshian system the accounting identity decomposes the total value of production. The Leontief system is based on a production activity and the Ghoshian on an allocation activity. Production and allocation are dual activities. The Ghoshian system does not ‘take the demand for granted’ as it has been criticized45 the same way the Leontief system does not take primary inputs for granted. Final demand is in equilibrium with primary inputs. When speaking about final demand in reality we are speaking about the ‘use’ as Augustinovics46 correctly states. Interindustry accounting does not suppose anything about elasticities.47 There is no demand or supply function in the interindustry accounting identities but value of final demand and primary inputs.

The analytical assumption of linearity in the Ghoshian model underlines the proportionality of industrial gross production of an industry to its own output allocation. Linearity is applied to dual activities, production and allocation. The primary inputs driven model provides a level and composition of industrial output given the level and composition of primary inputs and output allocation coefficients. The Ghoshian inverse is different than the Leontief inverse in all elements but the main diagonal.

xT ≡ iTX + iTH B = <x>-1X

x = [I-BT]-1HTi = UTHTi = [HU]Ti.

The differences in the causal relations, the underlying analytical assumptions and the multiplier matrices Z = [I-A]-1 and U = [I-B]-1 require further investigation of the two approaches.

Accounting identities provide the equality of gross output as a summation of total interindustry transactions and total final demand, or as total interindustry transactions and primary inputs. This implies value of total final demand equal to the value of primary inputs. Their decomposition is different. The difference in their decomposition does not disturb their equilibrium at a scalar aggregate level.

The final demand driven model is based on the input direct requirement coefficients A structure where the primary inputs driven model is based on the output direct allocation coefficients structure B. A production structure defines an allocation structure and vice versa. If linearity is applicable, it is applicable in both, production and allocation. The difficulty is accepting linearity itself. As we accept linearity in production automatically it is accepted in allocation. Assumption about production is not abandoned in a Ghoshian model48 because we base the analysis on its dual allocation activity of a linear system.

45 Oosterhaven (1988, op. cit., p. 207) 46 Ibid. 47 ibid. 48 ibid.


Table 8 Comparisons of Modelling Aspects

Leontief Ghoshian

Data Allocation of Total Demand Requirements of Total Production

Assumption Proportional Requirements Proportional Allocation

Solution Gross Output Gross Output

The data used and (linearity) assumption is the same in both approaches. The two models provide the same solution for gross output . [ ZY] i = x, and [ HU] T i = [ U] T[ H] Ti = x.

One needs to examine more carefully the differences in matrices Z and U. Matrix Z measures the total production of sector (i) necessary to deliver a unit of final demand of sector (j). It relates total production to final demand that is to a unit of product leaving the interindustry system at the end of the production process. Matrix U measures the total value of production that comes about in sector (j) per unit of primary input in sector (i), and therefore relates total production to primary inputs entering the interindustry system at the beginning of the production process.

Matrices Z and U have the same main diagonal. As multiplier matrices then provide unitary and uniform multipliers in the summation of their columns and rows respectively. These multipliers are different in both approaches as one may see in the numerical example. The traditional unitary and uniform multipliers are given as the summation of row and column elements in the inverse matrices respectively. The total result (sum) of unitary and uniform multipliers is the same in each activity but different in both activities as well as in its multiplier decomposition.

The above differences brought about the conclusion that production and allocation models provide different results. The difference in the above results is due to the fact that both models have similar types of causal element injection applied to a different inverse structure. These results are admissible whenever one evaluates the impact due to same cause of different matrix multipliers. The differences in the traditional multipliers and as well as the differences in the off main diagonal elements of Leontief and Ghoshian inverses is not admissible evidence that Leontief and Ghoshian models provide different results. At the same time the fact that both models provide the same solution for gross output is admissible evidence that provides the same result in a different approach.

1.2541 0.33000.2640 1.1221⎡

⎣ ⎢ ⎤

⎦ ⎥ 50 3001200 500⎡

⎣ ⎢ ⎤

⎦ ⎥ 11⎡

⎣ ⎢ ⎤

⎦ ⎥ =10002000⎡

⎣ ⎢ ⎤

⎦ ⎥

1.2541 0.13200.6601 1.1221⎡

⎣ ⎢ ⎤

⎦ ⎥ 150 500120 1280⎡

⎣ ⎢ ⎤

⎦ ⎥ 11⎡

⎣ ⎢ ⎤

⎦ ⎥ =10002000⎡

⎣ ⎢ ⎤

⎦ ⎥


Table 9 Production Interconnected Structure

Leontief Inverse Industry 1 Industry 2 Unitary

Multipliers Industry 1 1.2541 0.3300 1.5841 Industry 2 0.2640 1.1221 1.3861 Uniform Multipliers 1.5181 1.4521 2.9702

Table 10 Allocation Interconnected Structure

Ghoshian Inverse Industry 1 Industry 2 Unitary

Multipliers Industry 1 1.2541 0.6600 1.9141 Industry 2 0.1320 1.1221 1.2541 Uniform Multipliers 1.3861 1.7821 3.1682

4.2 Quasi-Inverses

Leontief and Ghoshian inverses provide the total interrelations among industrial sectors. The Taylor expansion provides the convergence of the geometric sequence of input coefficients in ‘round effects’ of the initial I, direct A, and indirect input requirements due to a final demand vector.49

I + A + A2 + A3 + ... + An = [I-A]-1 for n approaching ∞

The similar expansion of the initial I, direct allocation coefficient matrix B, and indirect allocation coefficients approach the Ghoshian inverse.

I + B + B2 + B3 + ... + Bn = [I-B]-1 for n approaching ∞

Whenever the requirement round effect matrices are postmultiplied to a vector of final demand vector then yield the intermediate demand required in order to satisfy the final and total demand y and x.

Ay + A2y + A3y + ... + Any = Xi

A similar expansion of the Ghoshian model has a similar meaning with economic interpretations50 given. The injection vectors y and h have the same total value iTy = iTh but different element decomposition. Matrices A and B may be transformed to each other given gross output.51 Therefore any round of one

49 Takayama (1985, ibid., p. 362) and Miller and Blair (1985, op. cit., pp. 22-24) 50 Miller and Blair (1985, op. cit., pp. 358-359) 51 Augustinovics (1970, op. cit. p. 256) and Miller and Blair (1985, op. cit., p. 360)


approach may be transformed into the other. Each round effect of the output coefficient matrix premultiplied to the row of primary inputs yields the row of intermediate input requirements necessary to comply with primary inputs h and satisfy gross production x.

hTB + hTB2 + hTB3 + ... + hTBn = iTX

The following parallel presentation in every round indicates the same total effect but different sectoral decomposition.

A final demand column of [350, 1700] requires total intermediate demand of [650, 300] given the input coefficients. Similarly, a primary input row of [650, 1400] necessitates a value of [350, 600] in interindustry requirements given the allocation coefficients. Taylor's expansion is not ‘extremely implausible’52 case limited to uneven sector growth in the allocation model. It is a clear indication of the symmetry between the two approaches. Knowing A one could evaluate intermediate demand Xi and having B determine intermediate input iTX.

Table 11 Sectoral Decomposition of Round Effects

Total Ind. 1 Ind. 2

Intermediate Demand - ‘Demand‘ Side 950 650 300

Interindustry Requirements - ’Supply‘ Side 950 350 600

Table 12 Final Demand and Primary Input Round Effects

Input Req. FD Round Effect PI Allocati on C. Round Effect

Total Round Effect

Ay = 0.1500 0.2500 350 477.50 hTB = 650 1400 0.1500 0.5000 237.50 395.00 632.50

0.2000 0.0500 1700 155.00 0.1000 0.0500 66.58%

A2y = 0.0725 0.0500 350 110.38 hTB2 = 650 1400 0.0725 0.1000 75.13 138.50 213.63

0.0400 0.0525 1700 103.25 0.0200 0.0525 22.49%

A3y = 0.0209 0.0206 350 42.37 hTB3 = 650 1400 0.0209 0.0413 25.12 44.49 69.61

0.0165 0.0126 1700 27.24 0.0083 0.0126 7.33%

A4y = 0.0073 0.0063 350 13.16 hTB4 = 650 1400 0.0073 0.0125 8.22 14.78 23.00

0.0050 0.0048 1700 9.84 0.0025 0.0048 2.42%

A5y = 0.0023 0.0021 350 4.43 hTB5 = 650 1400 0.0023 0.0043 2.71 4.85 7.56

0.0017 0.0015 1700 3.12 0.0009 0.0015 0.80%

A6y = 0.0008 0.0007 350 1.45 hTB6 = 650 1400 0.0008 0.0014 0.89 1.60 2.49

52 Oosterhaven (1988, op. cit., p. 207)


0.0006 0.0005 1700 1.04 0.0003 0.0005 0.26%

A7y = 0.0003 0.0002 350 0.48 hTB7 = 650 1400 0.0003 0.0005 0.29 0.53 0.82

0.0002 0.0002 1700 0.34 0.0001 0.0002 0.09%

A8y = 0.0001 0.0001 350 0.16 hTB8 = 650 1400 0.0001 0.0002 0.10 0.17 0.27

0.0001 0.0001 1700 0.11 0.0000 0.0001 0.03%

A9y = 0.0000 0.0000 350 0.05 hTB9 = 650 1400 0.0000 0.0000 0.03 0.06 0.09

0.0000 0.0000 1700 0.04 0.0000 0.0000 0.01%

A10y = 0.0000 0.0000 350 0.02 hTB10

=

650 1400 0.0000 0.0000 0.01 0.02 0.03

0.0000 0.0000 1700 0.01 0.0000 0.0000 0.00%

A11y = 0.0000 0.0000 350 0.01 hTB11

=

650 1400 0.0000 0.0000 0.00 0.01 0.01

0.0000 0.0000 1700 0.00 0.0000 0.0000 0.00%

A12y = 0.0000 0.0000 350 0.00 hTB12

=

650 1400 0.0000 0.0000 0.00 0.00 0.00

0.0000 0.0000 1700 0.00 0.0000 0.0000 0.00%

The symmetry between the Leontief53 and Ghoshian inverses allows one to move the analysis a step further examining the quasi-inverse54 in both systems and their role. The quasi-inverse has a great economic importance allowing the distinction between direct and indirect effects. The quasi-inverses of an interindustry system, the one based on input requirements and the other on output allocation, have the same main diagonal.

A + A2 + A3 + ... + An = A* B + B2 + B3 + ... + Bn = B*

The requirements without the initial requirement are given by the quasi-inverses.

and

Total requirements including the initial are the Leontief and Ghoshian inverses.

53 Goodwin, R. M. (1949) “The Multiplier as a Matrix” The Economic Journal, 59, Dec. pp. 537-555.

Samuelson, P. A. (1943) “A Fundamental Multiplier Identity” Econometrica, July, pp. 221-226. Chipman, J. S. (1950) “The Multi-Sector Multiplier” Econometrica, 18, Oct. pp. 355-374.

54 Wong, Y. K. (1954) “Some Mathematical Concepts for Linear Economic Models” in Morgenstern, Oscar (ed.) (1954) Economic Activity Analysis, J. Wiley & Sons, NY, pp. 283-339 Wong, Y. K. (1954) “Note on the Quasi-inverse of a square matrix: Its economic significance and an application.” In Morgenstern, Oscar (ed.) (1954) Economic Activity Analysis, J. Wiley & Sons, NY, pp. 315-321 Adamou, N. (op. cit., 1988) pp. 30-36.

A*=0.2541 0.33000.2640 0.1221⎡

⎣ ⎢ ⎤

⎦ ⎥ B*=

0.2541 0.66010.1320 0.1221⎡

⎣ ⎢ ⎤

⎦ ⎥


and

Weighting the initial ejection to final demand plus the intermediate demand in term of a Taylor expansion yields total demand.

Iy + Ay + A2y + A3y + ... + Any = x Weighting the initial ejection to primary inputs plus the intermediate input in terms of a Taylor expansion provides total production.

hTI + hTB + hTB2 + hTB3 + ... + hTBn = xT Input requirements premultiplied to gross output provide intermediate demand.

Ax = Xi

Output coefficients postmultiplied to gross output give intermediate requirements.

xTB = iTX = XTi

Since A is the matrix of direct requirements, A* is the matrix of total requirements, indirect requirement transactions are given by the matrix of their difference. Indirect requirement is the operation (matrix multiplication) of total requirement A* on direct requirement A.

A* - A = A* A

Indirect requirement transactions are given as a matrix difference or as a matrix product. The same holds for indirect allocations. Requirements are satisfied by sales in an equilibrium system.

B* - B = B* B

Indirect requirement transactions are defined as (A*-A) and the indirect allocation transactions are given as (B*-B). The left hand side of the equation illuminates the concept of the indirect transaction as a requirement or allocation. On the right hand side of the equation the interrelation of indirect to direct transactions are given in terms of the matrix multiplying operation. This is an operation of the total requirement row on the direct requirement column valid in production and allocation activities.

Intermediate demand Xi may be expressed either in terms of input coefficients related to gross output Ax or as the difference between gross and net demand x-Yi. Similarly, intermediate inputs iTX are given in terms of output coefficients xTB as well as a difference between total production and total primary inputs xT-iTH.

The input coefficient matrix is the operator on the vector of gross demand to intermediate demand.

Ax = x - Yi = [650, 300]T

The output coefficient matrix is the operator on the vector of gross production to intermediate input requirements.

I +A* =1.2541 0.33000.2640 1.1221⎡

⎣ ⎢ ⎤

⎦ ⎥ I +B* =

1.2541 0.66010.1320 1.1221⎡

⎣ ⎢ ⎤

⎦ ⎥


xTB = xT-iTH = [350, 600]

Since [I+A*][I-A] = I, and x=[I+A*]y, as well as [I+B*][I-B] = I, and xT=hT[I+B*], then A*y=x-y and hTB*=xT-hT. This implies that A* operates on the final demand vector in the same way that A operates on the vector of gross output.

A*y = Ax = [650, 300]T

Similarly, B* operates in the vector of primary inputs in the same way that B operates on the vector of total production.

hTB* = xTB = [350, 600]

The above result may be expanded decomposing final demand from a vector y to a matrix Y, and primary inputs from h to H. This would decompose intermediate demand and intermediate inputs to their respective indirect requirements.

A*Y = and HB* =

The summation of row elements of the A*Y matrix provides the intermediate demand [650, 300]T and the summation of column elements of HB* intermediate inputs [350, 600].

In a similar way one may allocate gross output to demand or requirement dimensions as:

(I+A*)Y = and H(I+B*) =

The summation of row elements of the (I+A*)Y matrix provides the total demand [1000, 2000]T and the summation of column elements of H(I+B*) total production [1000, 2000]. The differences of the above two groups of matrices provide the final demand and primary input matrices as:

[(I+A*)Y - A*Y] = and [H(I+B*) - HB*] =

Due to the above, Wong indicated that the essential matrix is not the Leontief inverse but the indirect requirement matrix (A*-A). One may extend the symmetry to (B*-B). For the indirect requirement matrices we have the following properties:

A*-A = A*A = AA* and B*-B = B*B = BB*.

The difference between total input requirements (A*) and direct input requirements (A) is the indirect input requirements matrix (A*-A). The matrix of the indirect requirements is given also as the operation of the

408.75 241.25159.74 140.26⎡

⎣ ⎢ ⎤

⎦ ⎥ 53.96 113.66296.04 486.34⎡

⎣ ⎢ ⎤

⎦ ⎥

458.75 541.251359.74 640.26⎡

⎣ ⎢ ⎤

⎦ ⎥ 203.96 233.66796.04 1766.34⎡

⎣ ⎢ ⎤

⎦ ⎥

50 3001200 500⎡

⎣ ⎢ ⎤

⎦ ⎥ 150 120500 1280⎡

⎣ ⎢ ⎤

⎦ ⎥


quasi inverse of the input coefficients on the direct input requirements. One may also evaluate the same result as the operation of the direct requirements on the quasi inverse of the input coefficients.

The difference between the quasi inverse of the allocation coefficients (B*) and direct allocation coefficients (B) defines the indirect allocation coefficients matrix. The indirect allocation coefficients are also defined by the post-multiplication and pre-multiplication of the quasi inverse of the allocation coefficients to the direct allocation coefficients.

4.3 Multiplier Overestimation and Weighted Decomposition

In interindustry analysis Leontief and Ghoshian inverses are related to a final demand and primary input ejection utilizing the ceteris paribus assumption. This assumption ‘slices’55 the respective inverse into rows or columns. The unitary multiplier assumes a unit final demand (or primary input) in a given sector and zero to all others. As a result the multiplier for each industry is based only on the column of the respective inverse. Figure 11 indicates vividly this situation for the Leontief inverse.

Figure 11 Unitary Multiplier Structure

The uniform multiplier on the other hand, assumes demand (or primary inputs) of a unit in all sectors. The result of this impact is given by the summation of the rows of the respective inverses. Figure 12 shows this case for the Leontief inverse.

The above situations do not present realistic assumptions of final demand and primary input distributions. The multiplier should take into account the actual composition of the total final demand or primary inputs

55 The ‘slicing’ idea in its graphical form is Dr. Panethymitakis' idea.

A *−A = A*A =AA*=0.1041 0.08000.0640 0.0721⎡

⎣ ⎢ ⎤

⎦ ⎥

B *−B =B *B= BB*=0.1041 0.16010.0320 0.0721⎡

⎣ ⎢ ⎤

⎦ ⎥

Z11 0 Z21 0 Zn1 0

0 0 Z 1n 0 0

0 Z 12 0 0 0

1 0 0

0 1 0

0 0 1


and not the ceteris paribus unitary or the uniformity assumptions. This weighted multiplier takes into account the entire inverse matrix and not only a row or a column of it for a given sector. Leontief multiplier provides measurement about the connectedness of production among all industries. Distribution matrices Yc and Ya may be used as weights for the Leontief inverse Z. Similarly, matrices Hc and Hb weight the measurement of interindustry connectedness of the allocation activity given by the Ghoshian inverse U.

Whenever one weights the Leontief inverse with the distribution of a final demand is able to evaluate the actual impact of a ‘demand’ driven model. This is given by a post-multiplication of a final demand distribution to the Leontief Inverse. As a result one may acquire the final demand driven weighted output multiplier Zyc=Zya.

The primary input driven weighted output multiplier hcU=hbU, calculated as a pre-multiplication of a primary input distribution to a Ghoshian inverse, is the same as the final demand driven weighted output multiplier.

Figure 12

Uniform Multiplier Structure

The same weighted output multiplier from the Leontief and the Ghoshian approaches is admissible evidence of their equivalencies. The important fact is that the weighted multiplier is substantially lower in magnitude than unitary and uniform multipliers of production and allocation approaches. The selection of an unbiased multiplier is important. The final demand-primary input weighted multiplier is unbiased because it weights properly the interconnected production to the actual distribution of the demand and the interconnected allocation to the distribution of primary inputs. As one may observe from Table 13, the overestimation of the unitary and uniform multipliers is a serious problem. This overestimation is a result of the injection assumption.

1.2541 0.33000.2640 1.1221⎡

⎣ ⎢ ⎤

⎦ ⎥ 0.17070.8293⎡

⎣ ⎢ ⎤

⎦ ⎥ =0.48780.9756⎡

⎣ ⎢ ⎤

⎦ ⎥

0.3171 0.6829[ ]1.2541 0.66000.1320 1.1221⎡

⎣ ⎢ ⎤

⎦ ⎥ = 0.4878 0.9756[ ]

Z11 Z 1n 0 0 0 0

0 0 0 0 Zn1 1

1 1

1 1 1

1 1 1

0 0 Z21

0


Multiplier decomposition is the second issue that has significant practical implications. One may derive decomposed multiplier premultiplying matrices Yc and Hc to respective Leontief and Ghoshian inverses. Tables 14 and 15 provide relative numerical examples.


Table 13 Multiplier Overestimation

Multipliers Unitary Leontief

Multipliers

Uniform Leontief

Multipliers

Unitary Ghoshian

Multipliers

Uniform Ghoshian

Multipliers

Final Demand and/or

Primary Inputs Distribution Multipliers

Industry 1 1.5841 1.5181 1.9141 1.3861 0.4878

Industry 2 1.3861 1.4521 1.2541 1.7821 0.9756 Overestimation Ind.1 Absolute 1.0963 1.0303 1.4263 0.8983 Ind. 1 Relative 3.25 3.11 3.92 2.84 1.00 Ind. 2 Absolute 0.4105 0.4765 0.2785 0.8065 Ind. 2 Relative 1.42 1.49 1.29 1.83 1.00

Table 14 Decomposed demand driven weighted output multiplier

Yc Z Multiplier Decomposition Multiplier 0.0244 0.1463 1.2541 0.3300 0.2238 0.2640 0.4878 0.5854 0.2439 0.2640 1.1221 0.6633 0.3123 0.9756

Table 15 Decomposed supply driven weighted output multiplier

Hc U Multiplier Decomposition 0.0732 0.0585 1.2541 0.6600 0.0995 0.11398 0.2439 0.6244 0.132 1.1221 0.3883 0.86160

Multiplier 0.4878 0.97558

4.4 Involved Determinants and Equivalent - Similarity Transformations

The interindustry structure is provided by the requirement coefficient A and the allocation coefficients B. These two matrices are based on the same information, X and x, and are derived using the same analytical assumption of linearity. For the derivation of these matrices adjusted total production is used, subtracting gross inventory depletion from total sectoral production.56 The structure of both matrices is given as follows.

and

A := X<x>-1 and B := <x>-1X

56 Miernyk, W. 1969 Input-Output Analysis, p.21.

aij :=Xijxj

bij :=Xij

xi


This indicates that interindustry transactions take the form either of the technological coefficients of production or the allocation coefficients.

X = A<x> = <x>B Matrices A and B are equivalent, related by a similarity transformation.57

A = <x> B <x>-1

B = <x>-1A <x> Matrices [I-A] and [I-B] have a distinct economic interpretation. Matrix [I-A] provides the structure of production and [I-B] the structure of allocation. Reading across the columns, an element of matrix [I-A] indicates the amount of commodity xj necessary to produce a unit of commodity xj. A negative element for

i≠j implies required input, a positive element indicates produced output. Reading across the row of matrix [I-B], a negative element indicates the distribution of output to other sectors, and the positive element (1-bii) = (1-aii) , located at the main diagonal, the produced output.

The rank of matrices A and B may be different, but not the rank of matrices [I-B] and [I-A] is the same. The rank of a matrix is the minimum number of dimensions in space required to depict the row or columns of a matrix as vectors. A matrix of a full rank implies that its rows or columns are linearly interdependent, and its determinant is not vanishing. A determinant is vanished if a row or a column is zero. It is not unusual to have a row or a column of zeros in matrices A or B. If the product of an industrial sector is final, for example shoes, its total production is allocated only to the final demand and not between industrial and final demand. Then, the relative row of B has only zero elements, but the relative column of A has nonnegative elements since inputs are required from another sector in order to produce shoes. Public services could be another example. In these cases, the diagonal element of production and allocation matrices are equal. A solution to the system exists if and only if production and allocation matrices are of full rank.

Similarity transformation interrelates production and allocation matrices. Using elementary operations we have

[I-A] = <x>[I-B]<x>-1 or

[I-B] = <x>-1[I-A]<x>

When we have two matrices of the same order, the determinant of their product is the product of their determinants. The production and allocation matrices have the same determinant,58 as

det(I-A) = det(I-B).

and

57 Equivalent transformations are defined by the expression

A=PBQ where P and Q are non-singular matrices of a proper dimensionality. Similarity transformations as well as congruence and orthogonal

transformations are used in interindustry modeling. For the definitions of the above and their use in interindustry modeling see: Stone, Richard, 1961, Input-Output and National Accounts, OEEC, pp. 87-88.

58 det (I-A) = (1-a11)(1-a22)-(a21a12) = (1-b11)(1-b22)-((x21/x1)(x12/x2)) =(1-b11)(1-b22)-((x21x2)(x12/x1)= (1-b11)(1-b22)-(b21 b12) = det (I-B).

det(I −A) =0.85 −0.25−0.20 0.95 = 0.7575 det(I −B) =

0.85 −0.50−0.10 0.95 = 0.7575


The meaning of the determinant is the value of net production as final demand or as the value of primary inputs payments.59 A similar relation exists also among the determinants of A and B,60 although the economic meaning of these determinants is not as clear as the det(I-A) and det(I-B).

det(A) = det(B)

and

A graphical interpretation of the equivalence of determinants (I-A) and (I-B) is presented later on in this section.

Production and allocation inverse matrices are similar.61

[I - A]-1 = <x>[I - B]-1<x>-1

[I - B]-1 = <x>-1[I - A]-1<x>

Total production is a function of primary inputs from an allocation of output point of view. Similarly, total production is a function of final demand from the production of output point of view. Primary inputs and final demand are major policy targets.

det(I-A)-1 = det(I-B)-1

and

Production and allocation determinants, det(I-A) & det(I-B), may be presented graphically in the net output space62 and in the gross output space.63 Net output may be seen either as net demand (final) or as net inputs (primary). Figures 13 and 14 present det(I-A) & det(I-B) respectively.

In Figure 13 we have each one of the vectors of the [I-A] production matrix.

Vectors (oa1) = and (oa2)= define the parallelogram

21yyyΟ , the determinant of the matrix [I-A]. The magnitude of this determinant specifies the final demand necessary for a gross output unit decomposed to its different types. Similarly, we have the graphical

59 For more detailed discussion of the structure of the linear demand driven models see:

Wong, Y. K., 1954, “Some Mathematical Concepts for Linear Economic Models”, in Oscar Morgenstern (ed.) Economic Activity, J. Wiley and Sons, N.Y., pp. 283-339.

60 det (A) = (a11)(a22)-(a21a12) = (X11/x1)(X22/x2)-((X21/x2)(X12/x1)) = (b11 b22)-((X21/x1)(x12/x2) = (b11)(b22)-(b21 b12) = det (B)

61 [I-A]-1 = {<x>[I-B]<x>-1}-1 = <x>-1)-1[I-B]-1<x>-1

62 Wong (op. cit.) 63 Dorfman, Solow & Samuelson (op. cit.)

det(A)=0.15 0.250.20 0.05 = −0.0425 det(B)=

0.15 0.500.10 0.05 = −0.0425

det(Z)=12541 0.33000.2640 1.1221 =1.32013 det(U) =

12541 0.66010.1320 1.1221 = 1.32013

1 − a11−a 21

⎛

⎝ ⎜ ⎞

⎠ =

0.85−0.20⎛

⎝ ⎜ ⎞

⎠

−a121 − a22⎛

⎝ ⎜ ⎞

⎠ =

−0.250.95

⎛

⎝ ⎜ ⎞

⎠


presentation of the [I-B], allocation matrix. Vector ob1 = = and ob2 = =

determining their own parallelogram 21hhhΟ .

Figure 13 Interindustry Systems in the Net Output Space

As one may observe, vectors (oy) and (oh), the summation vectors, have the same magnitude but different coordinates since final demand and primary input are decomposed in the aspects of net output space differently. The equivalence transformation allows one to transform the coordinates of one into the coordinates of the other.

The approach in Figure 13 depicts mathematically the vector space of [I-A] and [I-B]. At the same time Figure 14 represents the row space of [I-A] and [I-B].

Figure 14 represents graphically the rows of the linear systems (I-A)x=Yi and (I-BT)x=HTi. The solution vector OE is defined at the intersection of the lines defined by the equations of the linear systems. Each equation defines the intersection points and the slope of the lines. Table 16 provides this information in comparative terms.

1 − b11−b 21

⎛

⎝ ⎜ ⎞

⎠

0.85−0.10⎛

⎝ ⎜ ⎞

⎠

−b121− b22⎛

⎝ ⎜ ⎞

⎠ −0.550.95

⎛

⎝ ⎜ ⎞

⎠

Production System (I-A)x=Yi Net Product in Decomposed Final Demand Terms

Allocation System (I-B)ʹ′x=H ʹ′i Net Product in Decomposed Primary Input Terms

y 2, h 2

h

y

h 1, y 1

(1-a 11)=(1-b 11)

-a 21-b 21

(1-a 22)=(1-b 22)

-b 12

-a 12

y 2

y 1 h 1

h 2

ϑθ

h 2

h 1

y 1

y 2


One clearly may observe that distances OA, OB and OA v& OB vare defined as having the same denominators, the elements of the main diagonal of the respective matrices, while numerators are the respective values of sectoral final demand yj or primary input hi. The distances OC and OC vand OD and

OD v are defined having differences in both numerator and denominator. It is noticeable that as the symmetric position of their respective denominators. Likewise is the case of the numerators in the comparison of the slopes.

Figure 14 Gross Output Space

in the Production [(I-A)x=Yi] and Allocation [(I-BT)x=HTi] Systems

A

A ʹ′

B

Bʹ′

C Cʹ′

D

Dʹ′

ψ ψʹ′ φʹ′

φ

Production System (I-A)x=Yi Lines 1 (AD) & 2 (CB)

Allocation System (I-B)ʹ′x=H ʹ′i Lines 3 (Aʹ′Dʹ′)& 4 (Cʹ′Βʹ′)

Line 2

Line 1

Line 4

Line 3

x1

x2

xOA = y 1 /(1-a 11) OB = y 2 /(1-a 22) OC = -(y 2 /a 21) OD = -(y 1 /a 12) = a 12 /(1-a 11) = a 21 /(1-a 22)

OAʹ′ = h 1 /(1-b 11) OBʹ′ = h 2 /(1-b 22) OCʹ′ = -(h 2 /b 21) ODʹ′ = -(h 1 /b 12) ʹ′ = b 12 /(1-b 11) ʹ′ = b 21 /(1-b 22)

x =


Table 16 Comparative Examinations of Models in Figure 14

System (I-A)x=Yi (I-BT)x=HTi Comments

OA and OA v

Final demand of

industry 1 relative to net requirements of industry 1 from itself in order to satisfy a unit of its own

final demand

Primary input to industry 1 relative to what

industry 1 provides to itself in order to satisfy a unit of its own primary

input

Only Numerator (final demand y

1and

primary input h1

of industry 1) varies, while the denominator of

the main diagonal of both production &

allocation matrices is the same

(1-a11) = (1-b11)

OB and OB v

Final demand of

industry 2 relative to net requirements of industry 2 from itself in order to satisfy a unit of its own

final demand

Primary input to industry 2 relative to what

industry 2 provides to itself in order to satisfy a unit of its own primary

input

Only numerator varies while the denominator of



(1-a22) = (1-b22)

OC and OC v

Final demand of industry 2 relative to

what industry 1 purchases from industry

2.

Primary input of industry 2 relative to

what industry 1 sales to industry 2.

Both numerator and denominator vary

OD and OD v

Final demand of industry 1 relative to industry 2 purchases

from industry 1.

Primary input of industry 1 relative to industry 2 sales to

industry 1.

Numerator and

denominator vary

f and f v .

Requirements of

industry 2 from industry 1, relative to net

requirements of industry 1 from itself in order to satisfy a unit of its own

final demand

Sales of industry 1 to industry 2, relative to

what industry 1 provides to itself in order

to satisfy a unit of its own primary input




(1-a11) = (1-b11)

y11− a11

h11− b11

y21− a 22

h 2

1− b22

−y2a21

−h2b12

−y1a12

−h1b21

a121− a11

b211− b11


y and y v

Requirements of

industry 1 from industry 2, relative to net

requirements of industry 2 from itself in order to satisfy a unit of its own

final demand

Sales of industry 2 to industry 1, relative to

what industry 2 provides to itself in order

to satisfy a unit of its own primary input




(1-a22) = (1-b22)

4.5 Fundamental Interindustry Identity

Augustinovics64 defined the distribution matrices and as total content indicators.

Distribution matrix65 shows how a produced output unit in each sector is distributed to the share

of primary input to total production. Similarly, the matrix66 distributes an allocated output unit of each sector to the share of final demand to total demand. The sum of row elements of matrix

and the sum of column elements of matrix yields one.

One may relate the total content indicator to final demand matrix Y, and the total content

indicator to primary input matrix H. These two matrices are identical indicating the symmetry of the two approaches. This constitutes the fundamental interindustry identity.

64 Op. cit.. Augustinovics calls matrices [UY

b] & [H

aZ] as well as the weighted multiplier matrices [ZY

a] and [H

bU] total content indicator. The

analytical distinction of those matrices are examined by Adamou (op. cit. 1988).

65 Proof: From xT ≡ i

TX + i

TH and X= A<x>, as well as H= H

a<x> we have:

iT

A<x> + iT

Ha<x> = i

T<x> ⇔ i

T<x> - i

TA<x> = i

TH

a<x> ⇔ i

T - i

TA = i

T H

a

iT

[I-A] = iT

Ha ⇔ i

T = i

TH

a[I-A]

-1 ⇔ i

T = i

T[H

aZ]

66 Proof: In x ≡ Xi + Yi we substitute X= <x>B and Y= <x>Yb

having

<x>Bi + <x>Yb

i = <x>i ⇔ <x>i - <x>Bi = <x>Yb

i ⇔ <x>-1

<x>i - <x>-1

<x>Bi = <x>-1

<x>Yb

i

[I-B]i = Yb

i ⇔ i = [I-B]-1

Yb

i ⇔ i = [UYb

]i

a 21

1− a 22

b121− b22

→ UYb[ ] HaZ[ ]↓HaZ[ ]↓

→ UYb[ ]

→ UYb[ ] HaZ[ ]↓

UYb[ ]i =1.2541 0.66010.1320 1.1221⎡

⎣ ⎢ ⎤

⎦ ⎥ 0.05 0.300.60 0.25⎡

⎣ ⎢ ⎤

⎦ ⎥ 11⎡

⎣ ⎢ ⎤

⎦ ⎥ =0.4587 0.54130.6799 3.3201⎡

⎣ ⎢ ⎤

⎦ ⎥ 11⎡

⎣ ⎢ ⎤

⎦ ⎥ =11⎡

⎣ ⎢ ⎤

⎦ ⎥

iT HaZ[ ]= 1 1[ ]0.15 0.060.50 0.64⎡

⎣ ⎢ ⎤

⎦ ⎥ 1.2541 0.33000.2640 1.1221⎡

⎣ ⎢ ⎤

⎦ ⎥ = 1 1[ ]

0.2040 0.11680.7960 0.8832⎡

⎣ ⎢ ⎤

⎦ ⎥ = 1 1[ ]

HaZ[ ]↓→ UYb[ ]


One may provide an interpretation to the fundamental interindustry identity in the following two67 ways.

or as

The fundamental interindustry identity provides the column of total primary input decomposition68 whenever one sums its row elements.

Similarly whenever one adds the elements of its column we have the row of final demand decomposition.

67 For an interpretation of each of the different aspects of the fundamental interindustry identity derivation see Adamou N. & Gowdy J. (op. cit.

1991). 68 From interindustry matrix algebra one may derive the following:

HaZY[ ]≡ HUYb[ ]

(HaZ)Y[ ]≡ H(UYb )[ ] Ha(ZY)[ ]≡ (HU)Yb[ ]0.2040 0.11680.7960 0.8832⎡

⎣ ⎢ ⎤

⎦ ⎥ 50 3001200 500⎡

⎣ ⎢ ⎤

⎦ ⎥ =150 1201100 680⎡

⎣ ⎢ ⎤

⎦ ⎥

150 120500 1280⎡

⎣ ⎢ ⎤

⎦ ⎥ 0.4587 0.54130.6799 0.3201⎡

⎣ ⎢ ⎤

⎦ ⎥ =150 1201100 680⎡

⎣ ⎢ ⎤

⎦ ⎥

0.15 0.060.50 0.64⎡

⎣ ⎢ ⎤

⎦ ⎥ 459 5411360 640⎡

⎣ ⎢ ⎤

⎦ ⎥ =150 1201100 680⎡

⎣ ⎢ ⎤

⎦ ⎥

204 234796 1766⎡

⎣ ⎢ ⎤

⎦ ⎥ 0.05 0.300.60 0.25⎡

⎣ ⎢ ⎤

⎦ ⎥ =150 1201100 680⎡

⎣ ⎢ ⎤

⎦ ⎥

HaZYi[ ]= HUYbi[ ]⇒ H < x >−1 x[ ]= Hi[ ]⇒ Hi[ ]= Hi[ ]

HaZYi[ ]⇒ HUYbi[ ]⇒ Hi[ ]

150 1201100 680⎡

⎣ ⎢ ⎤

⎦ ⎥ 11⎡

⎣ ⎢ ⎤

⎦ ⎥ =2701780⎡

⎣ ⎢ ⎤

⎦ ⎥

iTHaZY[ ]= iTHUYb[ ]⇒ iTY[ ]= x < x > −1 Y[ ]⇒ iTY[ ]= iTY[ ]

iTY[ ]⇐ iTHaZY[ ]⇐ iTHUYb[ ]

1 1[ ]150 1201100 680⎡

⎣ ⎢ ⎤

⎦ ⎥ = 1250 800[ ]

x = [ZY]i Ha =H < x >−1 UYbi = i[HU]T i = x Yb =< x >−1 Y iTHaZ= iT


Total content indicators are independent of the ‘demand’ or ‘supply’ interindustry modeling approach. The column sum of the fundamental identity gives final demand.

4.6 Eigenvalues and Eigenvectors in ‘Demand’ and ‘Supply’ Interindustry Modelling

Eigenvalues indicate the growth of a sector, and eigenvectors the effect of such a growth rate.69 Therefore, each eigenvalue is associated to an eigenvector. Eigenvectors are important because when a matrix multiplies them, that matrix undergoes a transformation that alters its vector length, but it does not involve a rotation through any angle. Therefore, eigenvalues indicate the factor of expansion.70

Since production matrix A and technological coefficient matrix [I-A] are related linearly, the corresponding eigenvalues have the same relation.71 Let ζa to be the eigenvalue of the production matrix [I-A], and ξa the corresponding eigenvalue of the technological coefficient matrix A. Similarly let ζb to be the eigenvalue of the allocation matrix [I-B], and ξb be the eigenvalue of the output coefficients B.

ζa = 1 - ξa and ζb = 1 - ξb

The eigenvalues of the Leontief ωa and Ghoshian ωb inverses have a similar linear relationship to their corresponding quasi-inverses ϖa and ϖb.

ϖa = ωa - 1 and ϖb = ωb - 1

Eigenvalues of indirect impact matrices are linearly related to the eigenvalue of the quasi-inverse and the direct coefficient matrices. Let ϑa be the eigenvalue of the indirect

effect of the ‘demand’ driven model and ϑb the eigenvalue of the indirect effect of the ‘supply’ driven model.

ϑa = ϖa - ζa and ϑb = ϖb- ζb

An examination of the eigenvalues of all involved matrices in interindustry analysis in both ‘demand’ and ‘supply’ driven models indicates that both approaches have the same eigenvalue in their corresponding matrices as Table 17 indicates.

Table 17 Common Eigenvalues in the Leontief & Ghoshian Models

Technological Coefficients A 0.3291 -0.1291 Allocation Coefficients B Production Matrix (I-A) 0.6708 1.1291 Allocation Matrix (I-B) Leontief Inverse Z 1.4906 0.8856 Ghoshian Inverse U Quasi-Inverse A* 0.4905 -0.1144 Quasi-Inverse B* Indirect Coefficients A*A 0.1614 0.0147 Indirect Coefficients B*B

Woodbury72 showed that technological coefficient and production matrices have the same eigenvectors. The fact is that all involved matrices in each particular approach have the same eigenvectors. Eigenvectors

69 Miller and Blair, Op. cit.., pp. 350-351. 70 Hammer, A. G. 1970, Elementary Matrix Algebra for Psychologists and Social Scientists, Pergamon Press, Australia, pp. 76-80. This book

presents a good explanation of a factor analysis of a matrix. Structural analysis will benefit a lot by using factor analysis of the matrices involved in interindustry economics.

71 Woodbury, Max, A. 1952, “Characteristic roots of Input-Output Matrices”, in Oscar Morgenstern (ed.) Economic Activity Analysis, J. Wiley & Sons, NY, pp. 365-382.


of technological coefficients A, production matrix (I-A), Leontief inverse Z, production quasi-inverse A*, production indirect coefficients [A*A], are the same. Similarly eigenvectors of the allocation coefficient matrix B, allocation matrix (I-B), Ghoshian Inverse U, allocation quasi-inverse B*, and allocation indirect coefficients [B*B] are also the same. These two sets of eigenvectors are related, but this relationship is not in the scope of this paper. Although, can be seen that when the elements of one element is positive, the corresponding element of the similar matrix is negative, and vise versa.

Table 18 Eigenvectors

Leontief Ghoshian 0.8128 -0.6711 -0.9414 1.1590 0.5824 0.7493 -0.3373 -0.6470

72 Op. cit., p. 377.


List of Tables & Figures

Table 1 Interindustry Accounting (Concept Presentation) Table 1.2 Interindustry Accounting Matrix (Mathematical Formulations) Table 1.3 Interindustry Accounting (Numerical Example) Table 2 Typology of Descriptive Coefficients Table 3 Column Distributions (Input Coefficients) Figure 1 Input Distribution of a Sectoral Unit of Gross Production Figure 2 Distribution of different types of Final Demand units. Table 4 Row Distributions (Allocation coefficients) Figure 3 Distribution of a unit of Total Demand Figure 4 Industry Distributions of a unit in Primary Inputs & Total Production Table 5 Two-Dimensional Distributions of Intermediate Transactions, Final Demand and

Primary Inputs Figure 5 Two-dimensional interindustry distribution of an intermediate demand unit Figure 6 Two-dimensional industry-primary inputs distribution of a total primary input

unit Figure 7 Two-dimensional industry-final demand distribution of a total final demand unit Table 6 Two-dimensional distribution of the Gross Output Figure 8 Two-dimensional distribution of gross output to processing sectors Figure 9 Two-dimensional distribution of gross output to primary inputs Figure 10 Two-dimensional distribution of gross output to final demand Table 7 Descriptive Statistics in two-Dimensional Distributions Table 8 Comparison of Modelling Aspects Table 9 Production Interconnected Structure Table 10 Allocation Interconnected Structure Table 11 Sectoral Decomposition of Round Effects Table 12 Final Demand and Primary Input Round Effects Figure 11 Unitary Multiplier Structure Figure 12 Uniform Multiplier Structure Table 13 Multiplier Overestimation Table 14 Decomposed demand driven weighted output multiplier Table 15 Decomposed supply driven weighted output multiplier Figure 13 Determinants (I-A) and (I-B) in the Net Output Space Figure 14 Systems (I-A)x=Yi and (I-BT)x=HTi in the Gross Output Space Table 16 Comparative Examination of Figures 13 and 14 Table 17 Common Eigenvalues in the Leontief & Ghoshian Models Table 18 Leontief & Ghoshian Eigenvectors


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