Technical Note

Journal of Applied Input-Output Analysis, Vol. 4, 1997

Input-Output Characteristic Analysis

ByWang Tong*

Abstract
Characteristic of the input-output direct consumption coefficient matrix is a powerfultool for economic structure analysis, as well as a bridge to coordinate the technico-economic
structure and the social product structure. The social product structure, and the final productstructure, can be observed by looking at the characteristic vectors of this matrix; and,

therefore, the coordination relationship between one economic sector with others can be
discussed quantitatively.

Adjusting product structures or promoting technical progress can improve the effect ofsocial economic activities. By means of characteristic vectors we can point out the limitationof this effect.

Using China's 1981 and 1987 input-output tables and characteristic method, we canprovide a structure analysis of 1981 and 1987 China's economy.

L Basic Concepts
1.1. Economic Structure Vector
A vector, components of which are all non-negative numbers and the sum of allcomponents is equal to 1, is called economic structure vector. In another word, avector X = {Xx,,Xn) is a economic structure vector if Xi > 0 is real and

1=1

In the two dimensional case (when n = 2) all economic structure vectors form aneconomic structure of a right triangle AOB with two unit length on the right sides AO
and BO; and every straight line connecting the vortex point 0 and any point C locatedon the hypotenuse AB corresponds to an economic structure vector (ca cb). All
economic structure vectors fill this economic structure triangle, as shown in Figure 1.

Manuscript received February 10, 1994. Revised September 18, 1997.

* The author is with the Research Institute on Economic System and Management, No. 1 Nancaochang
Street, Beijing, 100035, China.

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Journal of Applied Input-Output Analysis, Vol. 4, 1997

Figure 1: Economic Structure Triangle

Let F = (FPF2) be a vector, and leti=then we call the vector /7=(/7'?/r') as a structurized vector corresponding to thevector F{Fi > 0) .
1.2. Input-Output Relationship of Structure TypeWe will discuss a national economy of n sectors. Let A be a matrix of direct
consumption coefficients. We can write the input-output relationship of price type forthis economy:

AX +Y = X,(i),nwhere Y = (YX,avectorx = (Xj,of9xn)'

JJ is a vector of final product,socialproduct.

Furtherlet

X = (XU

y = (y]9,XJ is9yn)'

be structurized vectors corresponding to YandandX

respectively. We call A the technico-economic structure, while y and X
are stmctures of final product and structure of social product respectively. In
this case we can write

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77

y = fyX = gxwhere j is the total value of final product and g is the total value of social product.
We can transform the input-output relationship of price type (1) into the input-output
relationship of structure typegAx + fy = gx
(2)

orAx + ty = x

where r=— is the ratio of the final product value to the social product value.
g1.3. Technico-Economic Structure
The social product structure and the final product structures are all one dimensional
structures, but the technico-economic structure is a two dimensional structure, which
is a basic structure of social economic activities. The input-output relationship ofstructure type describes the quantitative relation between the technico-economicstructure, the structure of final product and the structure of social product.
From formula (2) we can see that in certain technico-economic structures the final
product structure (multiplied by its ratio) can be uniquely determined by the social

product structure and inverse the social product structure —by the final productstructure (with its ratio); therefore the ratio of final product value —one of the mostimportant indicators of the effect of social economic activities— depends only on theabove three mentioned structures.

2. Methodology
2.1. Characteristic Matrix, Characteristic Equation, Characteristic Rootsand Characteristic Vectors

Let A be input-output matrix, / —the unit matrix and A — a parameter. ThenXI-A
is called the characteristic matrix, andXI - A = 0

(3)

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is

Journal of Applied Input-Output Analysis, Vol. 4, 1997

thecharacteristicequation;

thesolution of which

A = (A1,,Xn)arecharacteristic roots, and for every one of which there is a corresponding characteristic

vector R. of A, that satisfies

t = ^R,(i = l,,n)(4)

It is easy to prove that when l^jl < \,X = (I - A)'1Y = Y + fiX;AY = Y + (\ + Ai + % + ...)AY
n=0

(5)

Here we can see the economic meaning of characteristic roots: X = Y is the first
approximation of (5), i.e. the total product is the final one; but in order to produce

final product Y it needs A Y of productive material (the second approximation);at the same time, in order to produce productive material A Y, it needs again

A(AY) = A Y = A AY of productive material (the third approximation) and soon, and so forth. Thus, we view the productive process of social product X as ainfinite process, the characteristic root and its powers are the coefficients of"attenuation" in this process.
2.2. Main Characteristic of Technico-Economic Structure
According to the theorem of nonnegative matrix among characteristic roots, there is

only one real characteristic root Aj ,0 < /Ij < 1, and (A,1 < \ (J: = 25..., n) . Xx iscalled the main characteristic root of A and the structured vector R\ correspondingtoXx

is called the main characteristic vector of A

or simply the maincharacteristic.

2.3. Meaning of the Main Characteristic of Technico-Economic Structure
The main characteristic is a powerful tool of economic analysis and a bridge to coordinate the two dimensional technico-economic structure, and the one dimensionalsocial product structure. From (2) we can see that when a technico-economic main
characteristic is coincident with a social product structure, then the corresponding

final product structure is also coincident with this main characteristic, i.e. when X is
the main characteristic andAx = Xxx

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79

then we can write (2) as

Xx x + ry = xor(5)ry = (1 - Xx)x9 r = \-A}
In this case (when the social product or the final product structure coincides with themain characteristic) the ratio of final product value r depends only on the technicoeconomic structure, and we can see that the social product structure is co-ordinatedcompletely with the technico-economic structure. Obviously, only in this case is thesocial product structure coincident completely with the final product structure; and inother cases they may not be coincident with each other.

2.4. Auxiliary Characteristic of Technico-Economic Structure in the TwoSectors9 Economies
We can prove that for two sectors input-output relationship the main characteristic

vector R\ of which technico-economic structure must be located in the first quadrant,

another technico-economic characteristic vector Ri must be located in the second orthe fourth quadrant. This vector Ri is also called the auxiliary characteristic of twosectors technico-economic structure. Thus, in the case of two sectors, the main
characteristic is the unique direction, along which the social product structure and thefinal product structure can be coincident.

2.5. Coordinate Region
In the case of two sectors, we can observe the social product structure by using
technico-economic characteristics. Meanwhile, we can dispose the social productstructure X in a coordinate system in which the bases are the main characteristic Ri
and the auxiliary characteristic Ri, thus

It is not difficult to prove that the ratio of final product value is equal to(7)From this formula we can see that the main characteristic Ri separates thestructure triangle AOB into two parts: RjOA and R,OB (see Figure 2).

If R2 is located in the fourth quadrant, when X from Ri closes to R2

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Journal of Applied Input-Output Analysis, Vol. 4, 1997

oThe above five corresponding figures denote the location of X, Ri and R2 instructure triangle for different classification of 1981 China's economy. In thesefigures, the arrowhead represents the direction of T increase; the region with obliquelines is the non-coordinate region. From these figures, we can see that in 1981,
industry and commerce were coordinated with other economic sectors, transportation
was basically coordinated with other economic sectors, while agriculture, energy
were not well coordinated with other sectors.

In the fourth case of the classification (i.e. the transportation case) the main

characteristic vector R\ is just coincident with the social product structure, in thiscase the corresponding characteristic root Xx = 0.52 has obvious economic meaning,i.e. that the ratio of final product value is equal to the difference of unity and Xx,
r = 1-^ = 0.48.
In Table 2 is shown the effective limitation of adjusting product structure. In 1981

China, reducing the weight of industry in the whole economy or developing

agriculture, energy, transportation or commerce would promote the increase of V,
meanwhile the effect of rapidly developing agriculture and transportation would be
most obvious.

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83

Table 2: Effective Limitation of Structure Adjust12345maxr.73.61.57.71.62

minr.38.38.48.47.47

Table 3: Characteristics for 1987 Chinese Economy
Order

\K

1.31.15(.33.67)(.97-.03)(.26.76)(.20.80)

2.57

.05

(.75•25)(-.43.57)(.44.56)(.60.40)

3.54.25(.11.89)(.41-.59)(.02.98)(.06.94)

4.55.01(.03.97)(.68-.32)(.03.97)(.03.97)

5.55.01(.06.94)(.42-.58)(.06.94)(.06.94)hh>>

Xy4. Structure Analysis of 1987 Chinese Economy
For 1987 Chinese economy the total value of social product was 2291.5 billion RMB(current price). The final product value was 1039.1 billion RMB. The ratio of final
product value was V = 0.453. Using the 1987 Chinese input-output table we cancalculate all the necessary data for our structure analysis.
For the five mentioned above in B.7 classification (denoted by order numbers 1, 2,3, 4, 5) the corresponding characteristic roots Xx and X2 characteristic vectors Riand Ri, final product structure y and social product structure X are shown in theabove Table 3.

The following five corresponding figures denote the location of X, Ri and R2
in the structure triangle for different classification of the Chinese economy in 1987.
According to these figures we can conclude that: in 1987 in China, industry and theagriculture were coordinated with other economic sectors; the commerce andtransportation sectors were basically coordinated with other economic sectors and theenergy sector was not coordinated very well with other sectors.

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Journal of Applied Input-Output Analysis, Vol. 4, 1997

5. Comparison of 1981 and 1987 Chinese Economic Structure1. As shown in the above mentioned tables and figures, we can see thatcharacteristic vectors that correspond to five different sector's classification of1981 and 1987 Chinese economy keep almost the same relative locations.
From the viewpoint of characteristic analysis, we can conclude that thestructural change of the Chinese economy from 1981 to 1987 is slow and notvery obvious.
2. Due to successful agricultural reform, the situation of agriculture in the Chinese
national economy had been slightly improved between 1981 and 1987. The
agriculture sector changed from a non-coordinate sector in1981

to acoordinate sector in 1987. The transportation sector also changed its positionin the Chinese economic structure between 1981 and 1987.
3. China is a developing country, with the development of the Chinese economy
being based mainly on extensive production. At the same time, the macroefficiency of the Chinese economy was even worse from 1981 (then0.48) to 1987 (thenY =V = 0.45). China should focus its efforts especially on

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85

technological progress in order to reduce intermediate consumption.

References&» -»»«»• *~~°- "buuh House&»**
Hse ""Economy"

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