32 7 For the Student Inter-Industry Transactions and Input-Output Analysis Robert Dixon* Department of Economics The University of Melbourne 1. Introduction The goods and services which are produced by one firm and which will be further transformed by other firms as part of their production pro- cess are called ‘intermediate products’. This is to distinguish them from ‘final products’ which are the goods and services which are produced by any one firm and sold to ‘final users’ (con- sumers etc.). In any year, approximately 60 per cent of all sales by Australian producers are to other Australian producers. In other words the majority of sales of newly produced goods and services in Australia involve transactions be- tween producers, not between producers and households. Given the dominance of transac- tions involving intermediate products it is de- sirable to have a technique to study the produc- tion and sale of intermediate goods and services. This is the role of input-output analysis’ which provides a view of interdepen- dence caused by the passage of intermediate goods and services between producers. Note the use of the word ‘interdependence’. Inter- industry or input-output analysis emphasises the idea that the economy should be viewed as a complete system of interdependent indus- tries. Individual industries supply produced in- puts to each other, they compete for the supply of labour, natural resources and capital, they compete for sales in domestic markets and they compete in international trade. The implica- * I am grateful to John Creedy for helpful comments on earlier drafts of this article. tions of industrial interdependence, be it via sales (forward linkage) or purchases (back- wards linkage), are often crucial to the under- standing of the effects of changes in economic circumstances both on particular industries and on the economy as a whole. Inter-industry analysis entails not only the description but also the analysis of the interdependence which exists amongst industries or sectors of the economy and has two aims: (i) to measure and describe inter-industry transactions; and (ii) to assist in forecasting the direct and the indirect effects of exogenous shocks (such as policy changes) upon individual industries or sectors. The starting point for both description and analysis is the inter-industry flow table.* 2. The Flow (or Transactions) Table The inter-industry flow table for any year3 at- tempts to record the flows of goods and ser- vices occurring between industries or sectors, the sales of each industry or sector to the vari- ous categories of final demand and the amounts spent by each industry on wages, imported raw materials, gross operating surplus4 etc. A flow table is a rectangular array of data (that is, it is arranged in rows and columns) containing information on the industrial com- position of final demand (and thus the purchase and sale of final products) and also the pur- chases and sales of intermediate goods and ser- vices between industries. The flow table records various types of transaction including intra-industry transactions, where the buying and selling takes place between firms in the same industry or sector, and inter-industry transactions, where the buying and selling
Transcript
32 7
For the Student
Inter-Industry Transactions and Input-Output Analysis
Robert Dixon* Department of Economics The University of Melbourne
1. Introduction
The goods and services which are produced by one firm and which will be further transformed by other firms as part of their production pro- cess are called ‘intermediate products’. This is to distinguish them from ‘final products’ which are the goods and services which are produced by any one firm and sold to ‘final users’ (con- sumers etc.). In any year, approximately 60 per cent of all sales by Australian producers are to other Australian producers. In other words the majority of sales of newly produced goods and services in Australia involve transactions be- tween producers, not between producers and households. Given the dominance of transac- tions involving intermediate products it is de- sirable to have a technique to study the produc- tion and sale of intermediate goods and services. This is the role of input-output analysis’ which provides a view of interdepen- dence caused by the passage of intermediate goods and services between producers. Note the use of the word ‘interdependence’. Inter- industry or input-output analysis emphasises the idea that the economy should be viewed as a complete system of interdependent indus- tries. Individual industries supply produced in- puts to each other, they compete for the supply of labour, natural resources and capital, they compete for sales in domestic markets and they compete in international trade. The implica-
* I am grateful to John Creedy for helpful comments on earlier drafts of this article.
tions of industrial interdependence, be it via sales (forward linkage) or purchases (back- wards linkage), are often crucial to the under- standing of the effects of changes in economic circumstances both on particular industries and on the economy as a whole. Inter-industry analysis entails not only the description but also the analysis of the interdependence which exists amongst industries or sectors of the economy and has two aims: (i) to measure and describe inter-industry transactions; and (ii) to assist in forecasting the direct and the indirect effects of exogenous shocks (such as policy changes) upon individual industries or sectors.
The starting point for both description and analysis is the inter-industry flow table.*
2. The Flow (or Transactions) Table
The inter-industry flow table for any year3 at- tempts to record the flows of goods and ser- vices occurring between industries or sectors, the sales of each industry or sector to the vari- ous categories of final demand and the amounts spent by each industry on wages, imported raw materials, gross operating surplus4 etc.
A flow table is a rectangular array of data (that is, it is arranged in rows and columns) containing information on the industrial com- position of final demand (and thus the purchase and sale of final products) and also the pur- chases and sales of intermediate goods and ser- vices between industries. The flow table records various types of transaction including intra-industry transactions, where the buying and selling takes place between firms in the same industry or sector, and inter-industry transactions, where the buying and selling
328 The Australian Economic Review 3rd Quarter 1996
takes place between producers in different in- dustries or sectors.
The first step in constructing a flow table consists of classifying all the firms (or all of the operations of various firms) to one industry or another. Industries are usually defined in terms of homogeneity of product (for example, ice cream, aircraft, wooden furniture) but some- times they are defined according to homogene- ity of process (for example, mining, metal fab- rication). Sectors are broad groupings of industries (for example, the primary sector, which encompasses all types of agriculture, an- imal husbandry, fishing and hunting, forestry).
3. Rules for Constructing (and Reading) the Flow Table
The latest5 official input-output table we have for Australia is for 1989-90 and covers all en- terprises, grouped into 108 industries. It is compiled by the Australian Bureau of Statis- tics. For present purposes, I have aggregated that data into a small table with only nine sec- tors. However, even though the table we will use is highly aggregated, the activities of all en- terprisedfirms and industries should be re- corded in the table; that is, they will all have been classified to one or other sector. I have also aggregated all the types of final demands (demands for final products-this includes ex- ports, demands by households for consumer goods and services, etc.) into one category, simply called ‘final demand’.
Let’s look at Table 1 for Australia and begin with the top nine rows of data, each row corre- sponding to a different sector of the economy.
Reading from left to right, each row shows the output sold by each sector along the left- hand side of the table to each sector or destina- tion named across the top of the table. As we read across the first row we see at the extreme right-hand side the value of total sales (by def- inition this is the ‘total value of output’ for the sector) by all enterprises in the primary sector ($26250 million). In earlier columns, we see the destination of these sales; some ($1 1 550 million) went to final demand, some ($14700 million) went as intermediate goods to other sectors-for example, $10275 million went to
firms in the food, drink and tobacco sector as raw materials. Some $2399 million was sold to other firms in the primary sector. The more de- tailed tables published by the ABS show that these are, in major part, cereal grains used to feed sheep, cattle and other livestock. An imag- inary line which links row 1, column 1; row 2, column 2; row 3, column 3; etc. is called the ‘principal diagonal’. Entries along this princi- pal diagonal show the amount of intra-industry trade.
Each column, reading from top to bottom, shows the purchases made by each sector named along the top of the table from the sup- pliers along the left-hand side. That is to say, each column records the outlays by each indus- try or sector, on intermediate goods (raw ma- terials) and other factors of production.6 For example, take the construction sector; the en- tries in the seventh column show that it bought $48 million worth of inputs from the primary sector, $447 million from mining and so on down; it also paid out $12276 million in wages, salaries and supplements, its gross op- erating surplus was $13 074 million, it paid in- direct taxes of $1353 million and it imported raw materials worth $3793 million.
We also have a column telling us about the industry composition of final products pro- duced in that year in Australia. This is given to us under the heading ‘final demand’ and in- cludes production which was exported.
You will see from the heading that the table is described as a ‘matrix’. This means that the columns and rows appear in a particular order (indeed in this table, the same order) and that the ordering matters. As we shall see later we can manipulate tables of numbers set out in this fashion using a particular type of arithmetic called ‘matrix algebra’.
4. Interpreting the Australian Input-Output Flow Table
If we look at the heading to Table 1 we see that the table is recorded at ‘basic values’. This means that transactions are valued at the prices received by the producer rather than those paid by the buyers. Thus, the commodity flows are recorded at the value at which they leave the
Tab
le 1
A
ustr
alia
: Ind
ustr
y by
Ind
ustr
y Fl
ow M
atri
x 19
89-9
0 ($
mill
ion,
bas
ic v
alue
s, d
irect
allo
catio
n of
com
petin
g im
ports
)
TO
Food
Te
xtile
s O
ther
El
ectr
icity
Tr
ade
Fina
nce
and
Tota
l F
ind
Tota
l Fr
om
Prim
ary
Min
ing
etc.
et
c.
mon
ufac
. et
c.
Con
stru
c.
etc.
se
rvic
es
inte
rmed
iate
de
tnan
d su
pply
Prim
ary"
Min
ing
Food
, drin
k an
d to
bacc
o
Text
iles,
clo
thin
g an
d fo
otw
ear
Oth
er m
anuf
actu
ringb
Elec
trici
ty, g
as a
nd w
ater
Con
stru
ctio
n
Trad
e. tr
ansp
ort a
nd
com
mun
icat
ion
Fina
nce
and
serv
ices
Prim
ary
inpu
ts
Wag
es e
tc.
Gro
ss o
pera
ting
surp
lus
Indi
rect
taxe
s et
c.
Impo
rts
Tota
l
2 39
9 4
1071
76
2215
612 17
1937
I431
3 26
8
11 02
5
1281
9 14
26 2
50
58
2 10
8 88
12
2 16
6
537
181
2 03
8
1720
4 10
4
12 1
52
554
1523
27 2
41
1027
5 50
6
29
5
4 44
7 18
27
1849
2 34
6 52
8
393
110
8 15
3149
12
44
1520
68
2
4361
20
76
5448
I2
07
618
102
1075
16
23
3369
6 99
65
549
6 48
48
81
1 14
700
8179
26
07
447
52
167
1359
8
594
24
73
194
1831
8
340
432
4 79
I1
3 61
8 3
210
30 70
8 65
5 15
749
1136
6 10
913
7664
6
2462
45
29
333
1424
72
97
1769
7
82
26
68
623
1 792
28
12
1143
1 93
I 52
17
1296
5 12
529
5144
1
6865
11
22
3818
18
280
3678
0 72
218
2139
8 32
84
1227
6 39
950
8181
1 17
2528
21 I
76
8200
13
074
3140
0 54
869
1585
51
3 943
34
6 13
53
7 12
8 90
00
2432
5
15 58
9 36
5 3
793
4 74
5 72
76
3690
3
1234
07
2209
9 56
328
1282
88
2256
95
6.52
969
11 55
0
13 64
3
25 3
56
6 75
5
46 7
60
4 40
2
5351
6
76 8
47
153 4
77
I6 66
4
34 3
72
443
342
26 2
50
27 2
41
33 6
96
9 96
5
I23
407
22 0
99
56 3
28
128 2
88
225
695
172 5
28
1585
51
40 9
88
71 2
75
1096
310
Not
es: (
a) T
his
sect
or c
ompr
ises
agr
icul
ture
, for
estry
, and
fish
ing
and
hunt
ing.
(b
) Thi
s se
ctor
com
pris
es w
ood
prod
ucts
, pap
er a
nd p
rintin
g, c
hem
ical
s, pe
trole
um, m
etal
pro
duct
s, tr
ansp
ort e
quip
men
t. an
d m
achi
nery
.
Sour
ce: A
ustra
lian
Bur
eau
of S
tatis
tics
1994
, Aus
tral
ia17
Nat
iona
l Acc
ount
s: In
put-O
utpu
t Tu
bles
t 19
89-9
0, C
at. n
o. 5
209.
0, A
BS,
Can
berr
a
330 The Australian Economic Review 3rd Quarter 1996
producers before commodity taxes are charged and these taxes are recorded separately from the commodity flows where they arise. When this convention is adopted, the flows are said to be at basic value^.^
We also note in the heading that there is a ‘direct allocation of competing imports’. This means that all imports, no matter where they originate, no matter whether they are final products or imports of intermediate goods (raw materials) and regardless of whether they could be produced in Australia or not, are recorded in their own row of the table under the heading of (that is, in the row of) the ‘importing sector’. Thus the figure 34 372 in the bottom right-hand corner of Table 1 is imports of final goods and services. The figures which precede that one in the imports row are imports of raw materials.
The table records flows in $million. It mea- sures the money value of purchases and sales and not the quantities purchased and sold. (Ide- ally, we would like information on the quanti- ties produced and sold, not just the dollar value of transactions.)
For planning and forecasting purposes we use ‘coefficients’ which are calculated using
the data given in the flow table, not the flow table itself. These coefficients are called ‘input-output’ or ‘direct requirements coeffi- cients’.
5. Input-Output or Direct Requirements Coefficients
A very simple application of the input-output table is calculating inputs as a percentage of the output of an industry and using these percent- ages for estimating the input requirements for any given output of that industry. After an input-output flow table has been constructed for a given year, a table of input or input- output coefficients can be developed from it.
We begin by calculating the ‘direct require- ments coefficients’ (also known as ‘input- output coefficients’). By an ‘input-output co- efficient’ we mean the amount of inputs ‘di- rectly’ required from one industry in order to produce 1 dollar’s worth of output of another industry. The term ‘directly’ here refers to the fact that these are the dollar values of the pur- chases which firms in one industry actually make from the firms in another industry. The
Table 2 Australia: Direct Requirements Coefficients 1989-90
TO Food Textiles Other Electricitj) Trade Finance From Primary Mining etc. etc. manuf: etc. Construe. etc. andserv.
Notes: (a) Each element of Table 2 may be interpreted as follows: a coefficient at the intersection of row i (a typical row) and column j (a typical column) represents the amount of Australian production froin industry i which is directly required to produce I dollar’s worth of output of industry j . (b) The whole table (matrix) of direct requirements coefficients is often denoted by the symbol A; any one element is de- noted by the symbol a,,.
Source: Table I .
Dixon: Inter-Industry Transactions and Input-Output Analysis 33 1
‘direct requirements coefficients’ or ‘input- output coefficients’ are denoted by the symbol
We calculate the input-output coefficient by dividing all the entries (for inter-industry trans- actions) in any sector’s column of the flow table by the total value of sales (which, by ac- counting convention, is equal to the total value of purchases) for that sector as given in the flow table. Another way of putting this is to say that the input-output coefficients tell us about the allocation of each sector’s ‘budget’.
Let’s introduce some notation for these inter- mediate transactions. Let xfJ be the value of the flow (sales) from the industry in row i to the in- dustry in column j, and let XJ be the total value of purchases by the industry in column j. We define a,, to be the direct requirement coeffi- cient. It is calculated as a, = (x,lX,).
Table 2 reports the calculated values of the direct requirements coefficients (based on the data in the flow table). Each entry (element) of Table 2 is the calculated value of the input- output coefficients a,. They may be interpreted as follows: a coefficient at the intersection of row i (a typical row) and column j (a typical col- umn) represents the amount of Australian out- put from industry i directly required to produce 1 dollar’s worth of output of industryj. Input- output coefficients may be expressed either in monetary or physical terms. Our table is ex- pressed in decimal fractions of a dollar of total (direct) purchases (which, by construction, is equal to the total value of sales by the industry).
For any industry, its direct requirements co- efficients will be found listed down the rele- vant column of the table of ‘direct requirements coefficients’. These measure the amount of in- puts directly required (that is, purchased) from each industry in order to produce 1 dollar’s worth of output of a given industry. They tell us about the ‘input-mix’; that is, they tell us about the relative importance to each industry of the different inputs which are purchased by pro- ducers in that industry. As noted above, the ‘di- rect requirements coefficients’ are calculated by expressing the entries for the purchases of raw materials which appear in each column of the flow table as a proportion of that column’s total. For example, the second entry in the pn-
‘ 1
‘ I J ‘
mary column of Table 2 is 0.0002 and is calcu- lated by dividing 4 by 26250. The result, 0.0002, tells us that, for every 1 dollar’s worth of their sales, producers in the primary sector purchase 0.02 of 1 cent’s worth of inputs from producers in the ‘mining’ sector. The fifth entry in the primary column of Table 2 is 0.0844 and is calculated by dividing 2215 by 26 250. This coefficient, 0.0844, tells us that, for every 1 dollar’s worth of their sales, pro- ducers in the primary sector purchase 8.44 cents worth of inputs from producers in ‘other manufacturing’. All of the other entries in Table 2 are calculated in an analogous fashion.
6. Direct and Indirect Relationships between Sectors
Table 2 shows the direct purchases made by producers in a given industry from producers in all other industries within the processing sector for each dollar’s worth of current output. But an increase in final demand for the products of any industry (coming from households, for ex- ample) will lead to both direct and indirect in- creases in the output of all industries in the pro- cessing sector. If, for example, there is an increase in final demand for the products of in- dustry A, there will be direct increases in pur- chases from industries B, C , and so on. But in addition, when industry B sells more of its out- put to industry A, B’s demand for the products of industries C, D, etc. will likewise increase. And these effects will spread throughout the processing sector.
The importance of these indirect links, in practice, may be seen as follows: We can see from row 2 of column 1 of either Table 1 or Table 2 that firms in the primary sector them- selves purchase very little from firms in the mining sector. According to Table 2 the direct input-output coefficient is only 0.0002 (that is, about two one-hundredths of 1 cent for every dollar’s worth of sales by the primary sector). However, the (near) absence of a direct buying and selling link between firms in the two sec- tors should not be interpreted as meaning that an expansion in the level of activity in the pri- mary sector would not result in increased de- mand for intermediate products produced by
332 The Australian Economic Review 3rd Quarter 1996
the mining sector. Indeed, the indirect links be- tween the two sectors are quite strong as may be seen as follows. Firms in the primary sector demand (relatively) sizeable amounts of inputs from firms in the other manufacturing and the electricity, gas and water sectors. If we go to the columns for these two industries we find that they in turn depend relatively heavily upon the mining sector for supplies of intermediate goods. This means that despite the absence of a direct input-output link between primary and mining there are obvious (and sizeable) indi- rect links. This story about the existence of short or long indirect pathways between sectors could be told of any pair of sectors in the table. All sectors are linked directly and indirectly with each other although sometimes the indi- rect links are not as obvious as we have seen in the case of primary and mining.
At first sight then it would appear that if there were to be an expansion in primary pro- duction, this would have only a very small im- pact on the mining sector. But would we use these figures (the coefficients in Table 2) for forecasting the demand for intermediate goods and services which would result from an ex- pansion of the primary sector?
Consider the following. Imagine I wish to forecast the demand for commodities produced by the mining sector and suppose that primary sector sales to final users are forecast to in- crease by $1 billion. How much will this raise the denland for raw materials producedsup- plied by the minerals sector? 0.0002 x $1 bil- lion? NO, more (much more) than that, because as we have seen producers are also linked indi- rectly. To take just one example, although pro- ducers in the primary sector themselves pur- chase very little directly from producers in the mining sector, they do purchase a relatively high proportion of their inputs from ‘other manufacturing’ (the input-output coefficient in Table 2 is 0.0844) and producers in that sector (other manufacturing) buy a relatively large amount of inputs from mining. According to Table 2, producers in other manufacturing pur- chase 6.58 cents worth of inputs from mining for every dollar’s worth of their sales. This is one of many examples of an indirect link be- tween producers or industries. (Another obvi- ous example involving primary and mining is via the primary sector’s increased demand of electricity, gas and water.) In fact, every indus- try is directly or indirectly connected with
Table 3 Australia: Total Requirements Coeficients 1989-90
To Food Textiles Other Electricity Trade Finance Frotn Primary Mining etc. etc. manut etc. Construc. etc. and sen.
Notes: (a) Each element may be interpreted as follows: a coefficient at the intersection of row i (a typical row) and column j (a typical column) represents the output (Australian production) of industry i required directly and indirectly to produce 1 dollar of output absorbed by final demand (that is, final output) of industry j . (b) This table is the matrix [I - A]-’.
Source: Table 2.
Dixon: Inter-Industry Transactions and Input-Output Analysis 333
every other industry. Clearly, we need a way of measuring the links between sectors which takes into account not only the direct links but also the indirect links.
An integral part of input-output analysis is the construction of a table which shows the di- rect and indirect effects of changes in final de- mand. It shows the total expansion of output in all industries as a result of the delivery of 1 dol- lar’s worth of output outside the processing sector by any one industry. A ‘delivery outside the processing sector’ means a sale to house- holds, investors, foreign buyers, a government agency, or any other buyer included in the final demand sector. This table is called the table of Total Requirements Coefficients. These coeffi- cients tell us, for every 1 dollar’s worth of out- put delivered,for,final use from any one indus- try, how much output is required directly and indirectly from each industry in the economy. Table 3 reports these coefficients for Australia in 1989-90.
We use these data (the coefficients in Table 3) for forecasting demand, not the direct re- quirements coefficients given in Table 2. For example if I want to know the direct and indi- rect effect upon the mining sector of an expan- sion of sales to final demand by primary pro- ducers I would look for the figure in Table 3 which is at the intersection of the primary col- umn and the mining row. The figure there is 0.017. So my forecast of the impact of an ex- pansion in the primary sector upon the de- mand for raw materials produced by the min- ing sector would be 0.017 x $1 billion.* This will be nearer the truth than 0.0002 x $1 bil- lion.
An inspection of the table setting out the di- rect and the total requirements coefficients sug- gests that ‘electricity, gas and water’, ‘trade, transport and communication’, ‘finance and services’ and ‘other manufacturing’ are the key producers of intermediate products. (Their rows in Table 3 tend to have large numbers in them.) It follows that improvements in effi- ciency and cost reductions in these four indus- tries, in particular, will have considerable ‘downstream’ effects. It is not surprising then to notice that microeconomic reform efforts are concentrated in these four sectors.
7. The Calculation of Total Requirements Coefficients
The coefficients in Table 3 tell us, for any one industry (named at the top of the table), how much is required directly and indirectly from producers in each other industry (named down the left-hand side) in order for it (the one named at the top) to produce 1 dollar’s worth of sales for final users. They reveal to us interde- pendencies between firmstindustries which are not immediately apparent to the firms them- selves.
As noted earlier, these coefficients tell us, for every 1 dollar’s worth of output deliveredfor final use from any one industry, how much out- put is required, directly and indirectly, from each industry in the economy. How are they calculated? It’s a complicated story which we will approach in an indirect and intuitive man- ner.
Consider an economy which comprises only one industry which produces both a raw ma- terial and a finished good.
Let:
X = total sales Y = final demand a = the input-output coefficient (note that 0 <
a < 1)
Total sales (x) are made up of sales to final de- mand (0 and also intermediate sales (ax), so that:
X = a X + Y
This may be manipulated as follows:
X - a X = Y
X(l - a ) = Y
from which it follows that:
This last equation is an expression which gives the value of X as a function of Y, the value of
334 The Australian Economic Review 3rd Quarter 1996
final demand (the demand for final products). The value of the term ( I - a)-’ tells us the total amount by which X will have to increase to meet an increase in final demand by one unit.
We cannot use this type of arithmetic for the Australian table as we are dealing with a table which has a number of interdependent sectors, not just one self-contained sector. This means that in order to calculate the direct and indirect effects of changes to final demand, we will have to deal with equations expressing the rela- tionship between inputs and output for every sector, giving us a large simultaneous equation system to deal with. This is done using matrix algebra. In this article we will concern our- selves only with the interpretation and uses to be made of the tables which result from these manipulations. For students who are interested in this area I have included in an appendix a fairly elementary introduction to matrix alge- bra and its use in input-output analysis.
8. The Stability of Input-Output Coefficients
It is important to note that we use the data in Table 2 (that is the direct coefficients) to esti- mate the total requirements coefficients. This means that the accuracy and reliability of one will depend upon the accuracy and reliability of the other. The data used to calculate input- output coefficients may be up to five or six years old. Under what circumstances will ‘his- torical’ input-output coefficients still be valid? Remembering that input-output coefficients are just the inverse of the average product or ‘productivity’ of the input, it is possible to identify the circumstances under which the co- efficients will be constant or changing only very slowly.
First, since the reported coefficients are ‘value’ not ‘technical’ coefficients, both rela- tive prices and relative quantities must be con- stant. Second, industry ‘composition’ must be constant. Third, within each enterprise ‘techni- cal coefficients’ (that is, quantities of inputs per unit of output) must be constant (fixed). Technical coefficients are constant if (i) it is a feature of the technology (that is, an engineer- ing fact) that inputs must be used in fixed pro-
portions (they are complements) and are not able to be substituted for each other; or (ii) it is possible for inputs to be substituted for each other, but in fact relative input prices don’t vary very much if at all; or (iii) it is possible for inputs to be substituted for each other, but al- though relative input prices have been varying the degree of substitutability is so low that we observe no or only negligible changes in coef- ficients, and we must have constant returns to scale.
9. Summary
A high proportion of the production of Austra- lian firms is sold to other firms, not to final users such as households. One way in which economic theory has responded to this fact has been the development of input-output analysis. Input-output analysis allows us to describe and analyse these flows. It is also a useful tool for forecasting the effects of a change in the cir- cumstances of one industry upon all other in- dustries.
August 1996
Appendix: Matrix Algebra and Input-Output Analysis’
The interdependence of an economic system is identified in the inter-industry flow table which reports inter-industry transactions, sales to final demand and purchases of primary inputs.
The output of each industry is distributed as an input to some or all other industries and to final demand. This distribution can be ex- pressed as:
where
X i = total output of industry i x j j = output of industry i used by industry j Y, = final demand for output i
Conversely, for any industry the total output is equated to the value of inputs from other in- dustries plus the value added in production.
Dixon: Inter-Industry Transactions and Input-Output Analysis 335
X j = CxV+ VJ
where 5 = value added in industry j . The levels of inter-industry flows are ex-
plained by assuming that at least some part of the output of one industry required by another will vary with the level of activity in the second industry. The relationship between inputs and outputs may be assumed to be constant over a range of output such that:
( i = 1 . . . n ) I
where aV = the (direct) input-output coeffi- cient.
By combining the equations given above, the balance equation for each industry becomes:
x; - C U V X j = Y; 0’ = 1 . . . n) j
In matrix notation this system of n equations may be expressed as:
X - A X = Y
where X and Y are the column vectors of in- dustry outputs and industry final demand deliv- eries, respectively, and A is the matrix of (di- rect) input-output coefficients.
The above may be rewritten as:
[I - A]X = Y
where the symbol I denotes the identity matrix. The equation above can be solved to obtain
the relationship between levels of final demand for each industry and the required output of each industry.
X = [I - A]-’Y
In this form, the elements of the matrix [I - A]-’ are interdependence coefficients ex- pressing the total amount of output from indus- try i required per unit of final demand for the output of industry j . These total requirements coefficients can be calculated by matrix inver- sion with the aid of a computer. It is these inter- dependence coefficients which form the build- ing blocks for further analysis, since they
incorporate both the direct and the indirect re- lationships between industries.
Endnotes
1. Wassily Leontief is generally regarded to be the ‘inventor’ of modern input-output analysis. See Leontief (1 966, especially chapters 2 and 7) for his early articles on this subject. Leontief received the Nobel Prize in 1973 ‘For the de- velopment of the input-output method and for its application to important economic prob- lems’.
2. Good introductions to input-output analysis may be found in Chenery and Clark (1959), Miernyk (1967), Parmenter (1982) and Wolf- son (1978).
3. The Australian Bureau of Statistics (ABS) publishes input-output tables for Australia. At the time of writing (mid 1996) the most recent table is for the financial year 1989-90. That table was published in 1994.
4. Gross operating surplus is defined to be that which is left over out of total revenue once pur- chases of raw materials, payments of wages and salaries etc. have been allowed for. It is thus a residual or balancing item which ensures that total receipts (the total dollar value of sales) is equal to total disbursements or outlays.
5. As at August 1996.
6. As a matter of definition, the ‘value of pro- duction’ or ‘value added’ by any industry is equal to the total value sold; that is, the ‘value of output’ less the cost of raw materials and in- termediate products used up in the process of production.
7. Note that subsidies are treated as negative commodity taxes.
8. We should note that there are ‘economic’ links between producers which are not ‘cap- tured’ by this approach (the input-output ap- proach focuses only on the buying and selling of raw materials). Most obvious of these is the
336 The Australian Economic Review 3rd Quarter 1996
fact that as the primary sector and other sectors expand their sales not only will they buy more raw materials, they will also pay out more in- comes to households in the form of wages etc. and profits. The households will in turn de- mand more final goods and services and this will impact directly or indirectly upon the producers in all sectors (including the mining sector). Our method does not take this into ac- count. Also as the primary and other sectors ex- pand they will demand imported raw materials (likewise as household income grows the de- mand for imports of consumer goods will grow), this will affect the foreign currency markets and alter the exchange rate. This in turn may affect the level of sales by particular industries and thus the demand for mineral products. Furthermore, the increase in the level of economic activity might affect the demand for credit, for money and for other financial as- sets and this might affect interest rates which in turn might directly or indirectly affect the de- mand for mineral (and other) products.
9. Good introductions to matrix algebra in the context of input-output analysis may be found in O’Brien, Lewis and Guest (1989), Theil, Boot and Kloek (1965) and Yan (1969).
References
Australian Bureau of Statistics, Australian Na- tional Accounts: Input-Output Tables, Cat. no. 5209.0, ABS, Canberra, various years.
Chenery, H. B. & Clark, P. G. 1959, lnterin- dustry Economics, John Wiley & Sons, New York.
Leontief, W. 1966, Input-Output Economics, Oxford University Press, New York.
Miernyk, W. H. 1967, The Elements of Input- Output Analysis, Random House, New York.
O’Brien, D. T., Lewis, D. E. & Guest, J. F. 1989, Mathematics for Business and Eco- nomics, Harcourt Brace Jovanovich, Sydney.
Parmenter, B. R. 1982, ‘Inter-industry analy- sis: The ORANI model of Australia’s indus- trial structure’, in Industrial Economics: Australiun Studies, eds L. R. Webb & R. H. Allan, George Allen & Unwin, Sydney.
Theil, H., Boot, J. C. G. & Kloek, T. 1965, Op- erations Research and Quantitative Eco- nomics, McGraw-Hill, New York.
Wolfson, M. 1978, A Textbook of Economics, Methuen & Co., London.
Yan, C.-S. 1969, Introduction to Input-Output Economics, Holt, Rinehart & Winston, New York.
