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2 Foundations of Input–Output
Analysis
2.1 Introduction In this chapter we begin to explore the fundamental structure of the input–output model, the assumptions behind it, and also some of the simplest kinds of problems to which it is applied. Later chapters will examine the special features that are associated with regional models and some of the extensions that are necessary for particular kinds of problems – for example, in energy or environmental studies or as part of a broader system of social accounts. The mathematical structure of an input–output system consists of a set of n linear equations with n unknowns; therefore, matrix representations can readily be used. In this chapter we will start with more detailed algebraic statements of the fundamen- tal relationships and then go on to use matrix notation and manipulations more and more frequently. Appendix A contains a review of matrix algebra definitions and oper- ations that are essential for input–output models. While solutions to the input–output equation system, via an inverse matrix, are straightforward mathematically, we will discover that there are interesting economic interpretations to some of the algebraic results.
2.2 Notation and Fundamental Relationships
An input–output model is constructed from observed data for a particular economic area – a nation, a region (however defined), a state, etc. In the beginning, we will assume (for reasons that will become clear in the next chapter) that the economic area is a country. The economic activity in the area must be able to be separated into a number of segments or producing sectors. These may be industries in the usual sense (e.g., steel) or they may be much smaller categories (e.g., steel nails and spikes) or much larger ones (e.g., manufacturing). The necessary data are the flows of products from each of the sectors (as a producer/seller) to each of the sectors (as a purchaser/buyer); these interindustry flows, or transactions (or intersectoral flows – the terms industry and sector are often used interchangeably in input–output analysis) are measured for a
10
2.2 Notation and Fundamental Relationships 11
particular time period (usually a year) and in monetary terms – for example, the dollar value of steel sold to automobile manufacturers last year.^1 The exchanges of goods between sectors are, ultimately, sales and purchases of phys- ical goods – tons of steel bought by automobile manufacturers last year. In accounting for transactions between and among all sectors, it is possible in principle to record all exchanges either in physical or in monetary terms. While the physical measure is perhaps a better reflection of one sector’s use of another sector’s product, there are substantial measurement problems when sectors actually sell more than one good (a Cadillac CTS and a Ford Focus are distinctly different products with different prices; in physical units, however, both are cars). For these and other reasons, then, accounts are generally kept in monetary terms, even though this introduces problems due to changes in prices that do not reflect changes in the use of physical inputs. (In section 2.6 we will explore the implications of a data set in which transactions are expressed in physical units – for example, tons of steel sold to the automobile sector last year.) One essential set of data for an input–output model are monetary values of the transactions between pairs of sectors (from each sector i to each sector j ); these are usually designated as zij. Sector j ’s demand for inputs from other sectors during the year will have been related to the amount of goods produced by sector j over that same period. For example, the demand from the automobile sector for the output of the steel sector is very closely related to the output of automobiles, the demand for leather by the shoe-producing sector depends on the number of shoes being produced, etc. In addition, in any country there are sales to purchasers who are more external or exogenous to the industrial sectors that constitute the producers in the economy – for example, households, government, and foreign trade. The demands of these units – and hence the magnitudes of their purchases from each of the industrial sectors – are generally determined by considerations that are relatively unrelated to the amount being produced. For example, government demand for aircraft is related to broad changes in national policy, budget levels, or defense needs; consumer demand for small cars is related to gasoline availability, and so on. The demand of these external units, since it tends to be much more for goods to be used as such and not to be used as an input to an industrial production process, is generally referred to as final demand. Assume that the economy can be categorized into n sectors. If we denote by xi the total output (production) of sector i and by fi the total final demand for sector i ’s product, we may write a simple equation accounting for the way in which sector i distributes its product through sales to other sectors and to final demand:
xi = zi 1 + · · · + zij + · · · + zin + fi =
∑^ n
j = 1
zij + fi (2.1)
(^1) In Chapters 4 and 5 we will explore more recent distinctions between “commodities” and “industries” and see how these observations lead to alternative representations of the input–output model.
2.2 Notation and Fundamental Relationships 13
Table 2.1 Input–Output Table of Interindustry Flows of Goods
Buying Sector 1 · · · j · · · n
Selling Sector 1 z 11 · · · z 1 j · · · z 1 n .. .
.. .
.. .
.. . i zi 1 · · · zij · · · zin .. .
.. .
.. .
.. . n zn 1 · · · znj · · · znn
All of these primary inputs together are termed the value added in sector j. In addition, imported goods may be purchased as inputs by sector j. All of these inputs (value added and imports) are often lumped together as purchases from what is called the payments sector, whereas the z ’s on the right-hand side of (2.2) serve to record the purchases from the processing sector, the interindustry inputs (or intermediate inputs ). Since each equation in (2.2) includes the possibility of purchases by a sector of its own output as an input to production, these inter industry inputs include intra industry transactions as well. The magnitudes of these interindustry flows can be recorded in a table, with sectors of origin (producers) listed on the left and the same sectors, now destinations (purchasers), listed across the top. From the column point of view, these show each sector’s inputs; from the row point of view the figures are each sector’s outputs; hence the name input–output table. These figures are the core of input–output analysis.
2.2.1 Input–Output Transactions and National Accounts As was suggested by Table 1.1, an input–output transactions (flow) table, such as that shown in Table 2.1, constitutes part of a complete set of income and product accounts for an economy. To emphasize the other elements in a full set of accounts, we consider a small, two-sector economy. We present an expanded flow table for this extremely simple economy in Table 2.2. (We examine more of the details of a system of national accounts in Chapter 4 .) The component parts of the final demand vector for sectors 1 and 2 represent, respec- tively, consumer (household) purchases, purchases for (private) investment purposes, government (federal, state, and local) purchases, and sales abroad (exports). These are often grouped into domestic final demand ( C + I + G ) and foreign final demand (exports, E ). Then f 1 = c 1 + i 1 + g 1 + e 1 and similarly f 2 = c 2 + i 2 + g 2 + e 2. The component parts of the payments sector are payments by sectors 1 and 2 for employee compensation (labor services, l 1 and l 2 ) and for all other value-added items – for example, government services (paid for in taxes), capital (interest payments), land
14 Foundations of Input–Output Analysis
Table 2.2 Expanded Flow Table for a Two-Sector Economy
Processing Sectors Final Total 1 2 Demand Output ( x)
Processing 1 z 11 z 12 c 1 i 1 g 1 e 1 x 1 Sectors 2 z 21 z 22 c 2 i 2 g 2 e 2 x 2 Payments Value Added ( v ′) l 1 l 2 lC lI lG lE L Sectors n 1 n 2 nC nI nG nE N Imports m 1 m 2 mC mI mG mE M Total Outlays ( x ′) x 1 x 2 C I G E X
(rental payments), entrepreneurship (profit), and so on. Denote these other value-added payments by n 1 and n 2 ; then total value-added payments are v 1 = l 1 + n 1 , and v 2 = l 2 + n 2 , for the two sectors. Finally, assume that some (or perhaps all) sectors use imported goods in producing their outputs. One approach is to record these import amounts in an imports row in the payments sector as m 1 and m 2.^2 Total expenditures in the payments sector by sectors 1 and 2 are l 1 + n 1 + m 1 = v 1 + m 1 and l 2 + n 2 + m 2 = v 2 + m 2 , respectively. However, it is often the case that the exports part of the final demand column is expressed as net exports so that the sum of all final demands is equal to traditional definitions of gross domestic product, i.e., net of imports. In that case a distinction is often made between imports of goods that are also domestically produced (competitive imports) and those for which there is no domestic source (noncompetitive imports), and all the competitive imports in the imports row will have been netted out of the appropriate elements in a gross exports column. Under these circumstances it is possible for one or more elements in the net export column to be negative, if the value of imports of those goods exceeds the value of exports. (For example, if an economy exported d300 million of agricultural products last year but imported d350 million, the net exports figure for the agricultural sector would be d50 million.) Also, if the federal government sells more of a stockpiled item (e.g., wheat) than it buys, a negative entry in the government column of the final demand part of the table could result. If the negative number is large enough, it could swamp the other (positive) final demand purchases of that good, leaving a negative total final demand figure. The elements in the intersection of the value-added rows and the final demand columns represent payments by final consumers for labor services (for example, lC includes household payments for, say, domestic help; lG represents payments to
(^2) The treatment of imports in input–output accounts is much more complicated than this, but for the present we prefer to concentrate on the overall structure of a transactions table. We return to imports in section 2.3.4 below, and in more detail in Chapter 4.
16 Foundations of Input–Output Analysis
aircraft production last year – form the ratio of aluminum input to aircraft output, zij / xj [the units are ($/$)], and denote it by aij :
aij =
zij xj
value of aluminum bought by aircraft producers last year value of aircraft production last year
This ratio is called a technical coefficient; the terms input–output coefficient and direct input coefficient are also often used. For example, if z 14 = $300 and x 4 = $15, 000 (sector 4 used $300 of goods from sector 1 in producing $15,000 of sector 4 output), a 14 = z 14 / x 4 = $300/$15, 000 = 0.02. Since a 14 is actually $0.02/$1, the 0.02 is inter- preted as the “dollars’ worth of inputs from sector 1 per dollar’s worth of output of sector 4.” From (2.5), aijxj = zij. This is trivial algebra, but it presents the operational form in which the technical coefficients are used. In input–output analysis, once a set of observations has given us the result a 14 = 0.02, this technical coefficient is assumed to be unchanging in the sense that if one asked how much sector 4 would buy from sector 1 if sector 4 were to produce a total output ( x 4 ) of $45,000, the input–output answer would be z 14 = a 14 x 4 = (0.02)($45, 000) = $900 – when output of sector 4 is tripled, the input from sector 1 is tripled. The aij are viewed as measuring fixed relationships between a sector’s output and its inputs. Economies of scale in production are thus ignored; production in a Leontief system operates under what is known as constant returns to scale. In addition, input–output analysis requires that a sector use inputs in fixed pro- portions. Suppose, to continue the previous example, that sector 4 also buys inputs from sector 2, and that, for the period of observation, z 24 = $750. There- fore a 24 = z 24 / x 4 = $750/$15, 000 = 0.05. For x 4 = $15, 000, inputs from sector 1 and from sector 2 were used in the proportion p 12 = z 14 / z 24 = $300/$750 = 0.4. If x 4 were $45,000, z 24 would be (0.05)($45,000) = $2250; since z 14 = $900 for x 4 = $45, 000, the proportion between inputs from sector 1 and from sector 2 is $900/$2250 = 0.4, as before. This reflects the fact that
p 12 = z 14 / z 24 = a 14 x 4 / a 24 x 4 = a 14 / a 24 = 0.02/0.05 = 0.4;
the proportion is the ratio of the technical coefficients, and since the coefficients are fixed, then the input proportion is fixed. For the reader with some background in basic microeconomics, we can identify the form of production function inherent in the input–output system and compare it with that in the general neoclassical microeconomic approach. Production functions relate the amounts of inputs used by a sector to the maximum amount of output that could be produced by that sector with those inputs. An illustration is
xj = f ( z 1 j , z 2 j ,... , znj , vj , mj )
2.2 Notation and Fundamental Relationships 17
Using the definition of the technical coefficients in (2.5), we can see that in the Leontief model this becomes
xj =
z 1 j a 1 j
z 2 j a 2 j
znj anj
(This ignores, for the moment, the contributions of vj and mj .) A problem with this extremely simple formulation is that it is meaningless if a par- ticular input i is not used in production of j , since then aij = 0 and hence zij / aij is infinitely large. Thus, the more usual specification of the kind of production function that is embodied in the input–output model is
xj = min
z 1 j a 1 j
z 2 j a 2 j
znj anj
where min ( x , y , z ) denotes the smallest of the numbers x , y and z. In the input–output model, for those aij coefficients that are not zero, these ratios will all be the same, and equal to xj – from the fundamental definition of aij in (2.5). For those aij coefficients that are zero, the ratio zij / aij will be infinitely large and hence will be overlooked in the process of searching for the smallest among the ratios. This specification of the production function in the input–output model reflects the assumption of constant returns to scale; multiplication of z 1 j , z 2 j ,... , znj by any constant will multiply xj by the same constant. (Tripling all inputs will triple output; cutting inputs in half will halve output, etc.) For the reader who is acquainted with the economist’s production function geom- etry, we show four alternative representations of production functions in input space for a two-sector economy in Figure 2.1. A linear production function , depicted in Figure 2.1(a) assumes that output is a simple linear function of inputs, which means that the inputs are infinitely substitutable for each other for any level of output. The figure shows a set of isoquants (constant output lines) depicting higher and higher levels of output. A classical production function , depicted in Figure 2.1(b), also shows a set of iso- quants (now constant output curves) depicting higher and higher levels of output. For a given value of z 1 j in Figure 2.1(b), increasing z 2 j leads to increases in xj – intersec- tions with higher-value isoquants. In this case input substitution is also possible but not linearly, as indicated by the isoquants showing alternative input combinations that generate the same level of output. For example, moving rightward along a particular isoquant in Figure 2.1(b) can be accomplished by reducing the amount of input 2 and increasing the amount of input 1, or leftward by reducing z 1 j and increasing z 2 j. The shape of the isoquants in Figure 2.1(b) reflects two specific classical assumptions about how inputs are combined to produce outputs. The negative slopes of the isoquants represent the fact that as the amount of one input is decreased, the amount of the other input must be increased in order to maintain the level of production indicated by a specific isoquant. The fact that the curves bulge toward the origin (mathematically
2.2 Notation and Fundamental Relationships 19
two inputs are considered. From the Leontief production function, if z 1 j , z 2 j ,... , z ( n − 1 ) j were all doubled but znj were only increased by 50 percent (multiplied by 1.5), then the minimum of the new ratios would be znj / anj and the new output of sector j would be 50 percent larger. There would be excess and unused amounts of inputs from sectors 1, 2,.. ., ( n − 1 ). But since inputs are not free goods, sector j will not buy more from any sector than is needed for its production, and thus the input combinations chosen by sector j will lie along the ray as represented in Figure 2.1(c). In short, Leontief production functions require inputs in fixed proportions where a fixed amount of each input is required to produce one unit of output. Figure 2.1(d) shows an activity analysis production function , which is a generaliza- tion of the Leontief production function and is a piece-wise linear approximation of the classical production function. Each isoquant is represented by a connected set of line segments. Each segment is a linear production function applicable over a limited range of combinations of inputs to produce a given level of output. Once the notion of a set of fixed technical coefficients is accepted, (2.2) can be rewritten, replacing each zij on the right by aijxj :
x 1 = a 11 x 1 + · · · + a 1 ixi + · · · + a 1 nxn + f 1 .. . xi = ai 1 x 1 + · · · + aiixi + · · · + ainxn + fi (2.6) .. . xn = an 1 x 1 + · · · + anixi + · · · + annxn + fn
These equations serve to make explicit the dependence of interindustry flows on the total outputs of each sector. They also bring us closer to the form needed in input– output analysis , in which the following kind of question is asked: If the demands of the exogenous sectors were forecast to be some specific amounts next year, how much output from each of the sectors would be necessary to supply these final demands? From the point of view of this equation, the f 1 ,... , fn are known numbers, the aij are known coefficients, and the x 1 ,... , xn are to be found. Therefore, bringing all x terms to the left,
x 1 − a 11 x 1 − · · · − a 1 ixi − · · · − a 1 nxn = f 1 .. . xi − ai 1 x 1 − · · · − aiixi − · · · − ainxn = fi .. . xn − an 1 x 1 − · · · − anixi − · · · − annxn = fn
20 Foundations of Input–Output Analysis
and, grouping the x 1 together in the first equation, the x 2 in the second, and so on,
( 1 − a 11 ) x 1 − · · · − a 1 ixi − · · · − a 1 nxn = f 1 .. . − ai 1 x 1 − · · · + ( 1 − aii ) xi − · · · − ainxn = fi (2.7) .. . − an 1 x 1 − · · · − anixi − · · · + ( 1 − ann ) xn = fn
These relationships can be represented compactly in matrix form. In matrix algebra notation, a “hat” over a vector denotes a diagonal matrix with the elements of the
vector along the main diagonal, so, for example, x ˆ =
x 1 · · · 0 .. .
0 · · · xn
. From the basic
definition of an inverse, ( x ˆ)( x ˆ)−^1 = I , it follows that x ˆ−^1 =
1 / x 1 · · · 0 .. .
0 · · · 1 / xn
. Also,
postmultiplication of a matrix, M , by a diagonal matrix, d ˆ, creates a matrix in which each element in column j of M is multiplied by dj in d ˆ (Appendix A, section A.7). Therefore the n × n matrix of technical coefficients can be represented as
A = Zx ˆ−^1 (2.8)
Using the definitions in (2.3) and (2.8), the matrix expression for (2.6) is
x = Ax + f (2.9)
Let I be the n × n identity matrix – ones on the main diagonal and zeros elsewhere;
I =
so then ( I − A ) =
( 1 − a 11 ) − a 12 · · · − a 1 n − a 21 ( 1 − a 22 ) · · · − a 2 n .. .
− an 1 − an 2 · · · ( 1 − ann )
Then the complete n × n system shown in (2.7) is just^4
( I − A ) x = f (2.10)
For a given set of f ’s, this is a set of n linear equations in the n unknowns, x 1 , x 2 ,... , xn and hence it may or may not be possible to find a unique solution. In fact, whether or
(^4) This is parallel to the form Ax = b that is usually used to denote a set of linear equations. The difference is purely notational; since it is standard in input–output analysis to define the technical coefficients matrix as A , then the matrix of coefficients in the input–output equation system becomes ( I − A ). Similarly, convention is responsible for denoting the right-hand sides of the input–output equations by f (for final demand) instead of b.
22 Foundations of Input–Output Analysis
Table 2.3 Flows ( zij ) for the Hypothetical Example
To Processing Sectors 1 2 Final Demand ( fi ) Total Output ( xi )
From 1 150 500 350 1000 Processing Sectors 2 200 100 1700 2000 Payments Sector 650 1400 1100 3150 Total Outlays ( xi ) 1000 2000 3150 6150
Table 2.4 Technical Coefficients (the A Matrix) for the Hypothetical Example
Sector 1 Sector 2 (Agriculture) (Manufacturing)
Sector 1 .15. (Agriculture) Sector 2 .20. (Manufacturing)
manufactures are needed as intermediate ingredients. These are, of course, only the inputs needed from other producing sectors; there will be inputs of a more “nonpro- duced” nature as well, such as labor, from the payments sectors. For an analysis of interrelationships among productive sectors, these are not of major importance. We can now ask the question: If final demand for agriculture output were to increase to $600 next year and that for manufactures were to decrease to $1500 – for example, because of changes in government spending, consumers’ tastes, and so on – how much total output from the two sectors would be necessary in order to meet this new demand?
We denote this new demand as f new^ =
[
f (^) 1 new f (^) 2 new
]
[
]
. In the year of observation,
when f =
[
]
, we saw that x =
[
]
, precisely because, in producing to sat-
isfy final demands, each sector must also produce to satisfy the demands for inputs into the processes of production themselves. Now we are asking, for f (^) 1 new = 600 and
f (^) 2 new = 1500, what are the elements of x new^ =
[
xnew 1 xnew 2
]
? To satisfy the demands, xnew 1
can be no less than $600 and x 2 new no less than $1500. These would be the necessary outputs – the “direct effects” – if neither product were used in production and all output were directly available for final demand. But since both products serve as inputs, in a manner that is reflected in the technical coefficients of Table 2.4, it seems clear that in the end, more than $600 worth of agriculture goods and more than $1500 worth of
2.3 An Illustration of Input–Output Calculations 23
manufactures will have to have been produced in order to meet the new final demands. That is, there will be “indirect effects” as well. Both of these effects are captured in the input–output model. In the 2 × 2 case, | I − A | = ( 1 − a 11 )( 1 − a 22 ) − a 12 a 21 (Appendix A) and
adj( I − A ) =
[
( 1 − a 22 ) a 12 a 21 ( 1 − a 11 )
]
For this example, A =
[
]
so ( I − A ) =
[
]
; hence | I − A | =
0.7575 "= 0 and we know that L = ( I − A )−^1 can be found. Here we have
L =
[
]
Assuming that technology (as represented in A ), does not change, the needed total outputs caused by f new^ are then found as in (2.11):
x new^ = Lf new^ =
[
] [
]
[
]
These values – x 1 new = $1247.52 and x 2 new = $1841.58 – are one measure of the impact on the economy of the new final demands.^5 With this result for x new , it is straightforward to examine the changes in all elements in the interindustry flows table (as in Table 2.3) caused by f new. From the definition of coefficients in (2.8), Z = A x ˆ. With a constant A matrix and new outputs, x new , we find Z new^ = A x ˆ new^ =
[
]
; along with f new^ =
[
]
, we have the results shown in Table 2.5. The elements in the Payments Sector are found as the difference between new total outputs (total outlays) and new total interindustry inputs for each sector. (For the exam- ple we assume no change in payments sector transactions with final demand.) Notice that sector 1’s purchases are larger (reflecting an increase in final demand for that sector) and sector 2’s purchases are smaller (reflecting smaller demand for that sector). The input–output model allows us to deal equally easily with changes in demands and outputs instead of levels. Here and throughout, we use superscripts “0” to represent the initial (base year) situation and “1” for values of variables after the change in demands (instead of “ new ” as we did above). Assuming that technology is unchanged means A^0 = A^1 = A and L^0 = L^1 = L , so x^0 = Lf^0 and x^1 = Lf^1 ; letting 1 x = x^1 − x^0 and
(^5) Here xnew 1 and xnew 2 are shown to two decimals for comparison with results from an alternative approach in section 2.3.2. These x new^ values reflect computer calculations carried out with more than four significant digits and hence often will (as here) differ (to the right of the decimal point) from what the reader will produce with a hand calculator using the four-digit elements shown for A. In any actual analysis, such detail might be questionable because of the much less accurate data from which the technical coefficients are derived (compare the figures in Table 2.3).
2.3 An Illustration of Input–Output Calculations 25
To continue with the numerical example, suppose that ec 1 = 0.30 and ec 2 = 0. give the dollars’ worth of labor inputs per dollar’s worth of output of the two sectors. (We will examine the role of labor inputs and household consumption in an input–output model in some detail in section 2.5, below.) Then
ε = ˆ e ′ c x^1 =
[
] [
]
[
]
This indicates the values of labor inputs purchased by the two sectors. If, additionally, we have an occupation-by-industry matrix, P , where pij is the pro- portion of sector j employment that is in occupation i , then ˜ε = P εˆ gives a matrix of employment by sector by occupation type. For example, with k occupation types and two sectors,
P =
p 11 p 12 .. .
pk 1 pk 2
and
ε ˜ = P εˆ =
p 11 ec 1 x^11 p 12 ec 2 x^12 .. .
pk 1 ec 1 x^11 pk 2 ec 2 x^12
Column sums would give total labor use by sector; row sums give total employment of a particular occupational category across all sectors. (The vector P ε shows employment by occupational category, aggregated across all sectors.) Suppose that our economy has three occupational groups: (1) engineers, (2) bankers and (3) farmers, and
P =
(For example, this says that 40 percent of the agricultural labor force is farmers; 80 percent of manufacturing labor force is made up of engineers, etc.) Then
˜ε = P εˆ =
[
]
Column sums of ε˜ are 374.26 and 460.40, as expected (the elements of ε). Row sums give the economy-wide (across both sectors) employment of engineers, farmers and bankers, respectively. If sectoral disaggregation is not necessary, then
P ε =
[
]
gives employment by occupational type, across sectors.
26 Foundations of Input–Output Analysis
A wide variety of such conversion coefficients vectors (as in e ′ c ) or matrices (as in P ) is possible. For example, in arid regions, water-use coefficients, w ′ c =
[
wc 1 wc 2
]
could be used in w ′ c x ′^ to assess the water consumption associated with new outputs generated by new final demands. We explore these kinds of alternative impacts again in Chapter 6 on input–output multipliers, and in Chapters 9 and 10 , some of the energy and environmental repercussions of final demand impacts are discussed in detail.
2.3.2 Numerical Example: Hypothetical Figures – Approach II Consider the same economy, whose 2 × 2 technical coefficients matrix is given in
Table 2.4 and for which the projected f^1 vector is
[
]
. We can examine the question
of outputs necessary to satisfy this final demand in a more intuitive way that is less mechanical than finding elements in an inverse matrix.
- Initially, it is clear that agriculture needs to produce $600 and manufacturing, $1500. If the sectors are going to meet the new final demands, they could not get away with producing less than these amounts.
- However, to produce $600, agriculture needs, as inputs to that productive process, (0.15)($600) = $90 from itself and (0.20)($600) = $120 from manufacturing. These figures come from the coefficients in column 1 of the A matrix – the production recipe for agriculture. Similarly, to produce its $1500, manufacturing will have to buy (0.25)($1500) = $375 from agriculture and (0.05)($1500) = $75 from itself. Thus agriculture must, in fact, produce the $600 noted in 1, above, plus another $( 90 + 375 ) = $465 more, to satisfy the needs for inputs that it has itself and also that come from manufacturing. Similarly, manufacturing will have to produce an additional $( 120 + 75 ) = $195 to satisfy its own need plus that of agriculture for inputs to produce the “original” $600 and $1500.
- In item 2, above, we found the interindustry needs that resulted from production of $600 in agriculture and $1500 in manufacturing. These were $465 and $195, respectively. But now we realize that this “extra” production, above the $600 and $1500, will also generate interindustry needs – in order to engage in the produc- tion of $465, agriculture will need (0.15)($465) = $69.75 from itself and (0.20) ($465) = $93 from manufacturing. Similarly, manufacturing will now additionally need (.025)($195) = $48.75 from agriculture and (0.05)($195) = $9.75 from itself. The total new demands for the two sectors are thus $(69.75 + 48.75) = $118.50 and $( 93 + 9.75) = $102.75.
- At this point we realize that it is necessary to treat the additional $118.50 for agri- culture and $102.75 for manufacturing in the same fashion as the $465 and $195 in item 3. Hence we find additional required outputs of $43.46 and $28.84 from the two sectors.
- Continuing in this way, we find that eventually the numbers become so small that they can be ignored (less than $0.005).
28 Foundations of Input–Output Analysis
Looking at the first product on the right, the new final demand of $600 for agriculture output is multiplied by 1.2541. This can be thought of as (1 + 0.2541)(600). The (1)(600) reflects the fact that the $600 new agriculture demand must be met by producing $600 more agriculture output. The additional (0.2541)(600) captures the additional agriculture output required because this output is also used as an input to production activity in both agriculture and also manufacturing. Similarly, from (2.13),
x^12 = (0.2640)( 600 ) + (1.1221)( 1500 )
and the same logic explains why the coefficient (1.1221) relating manufacturing output to new final demand for manufacturing goods, $1500, must be greater than 1. We examine why both of the diagonal elements in L will be greater than 1 in the two-sector case. (A more complicated derivation can be used for the general n -sector input–output model, and it is also apparent from the power series discussion in section 2.4.) For this 2 × 2 example, as we saw in section 2.3.1, above,
L =
[
l 11 l 12 l 21 l 22
]
| I − A |
[adj( I − A )]
( 1 − a 11 )( 1 − a 22 ) − a 12 a 21
[
( 1 − a 22 ) a 12 a 21 ( 1 − a 11 )
]
So, for example,
l 11 =
( 1 − a 22 ) ( 1 − a 22 )
[
( 1 − a 11 ) − (^) ( a 112 − aa^2122 )
] =
[
a 11 + (^) ( a 112 − aa 2221 )
]
Assuming that ( 1 − a 22 ) > 0, l 11 > 1 if the denominator on the right-hand side is less than 1, which it will be when a 11 > 0 and/or a 12 a 21 > 0 – since ( 1 − a 22 ) > 0. Similar reasoning shows that l 22 = ( 1 − a 11 )/ | I − A | > 1 under similar reasonable conditions on the aij. Whether or not the off-diagonal elements are larger than 1 depends entirely on the sizes of a 12 and a 21 , relative to | I − A |. In most actual input–output tables, with a rather detailed breakdown of sectors, the off-diagonal elements in L will be less than 1, as in (2.13). However, for example, if a 21 in Table 2.4 had been 0.70 instead of 0.20, so that the coefficients matrix had been
A =
[
]
then L =
[
]
Notice that a coefficient as large as a 21 = 0.7 – which says that there is 70 cents’ worth of sector 2 output in a dollar’s worth of sector 1 output – is not likely to be seen
2.3 An Illustration of Input–Output Calculations 29
Table 2.7 The 2003 US Domestic Direct Requirements Matrix, A
Sector 1 2 3 4 5 6 7 1 Agriculture .2008 .0000 .0011 .0338 .0001 .0018. 2 Mining .0010 .0658 .0035 .0219 .0151 .0001. 3 Construction .0034 .0002 .0012 .0021 .0035 .0071. 4 Manufacturing .1247 .0684 .1801 .2319 .0339 .0414. 5 Trade, Transportation & Utilities
.0855 .0529 .0914 .0952 .0645 .0315.
6 Services .0897 .1668 .1332 .1255 .1647 .2712. 7 Other .0093 .0129 .0095 .0197 .0190 .0184.
Table 2.8 The 2003 US Domestic Total Requirements Matrix, L = ( I − A )−^1
Sector 1 2 3 4 5 6 7 1 Agriculture 1.2616 .0058 .0131 .0576 .0037 .0069. 2 Mining .0093 1.0748 .0122 .0343 .0193 .0033. 3 Construction .0075 .0034 1.0047 .0064 .0065 .0111. 4 Manufacturing .2292 .1192 .2615 1.3419 .0692 .0856. 5 Trade, Transportation .1493 .0850 .1371 .1563 1.0887 .0598. & Utilities 6 Services .2383 .2931 .2700 .2918 .2712 1.4116. 7 Other .0243 .0239 .0231 .0367 .0280 .0297 1.
often in real tables. The sizes of the between-sector technical coefficients, aij ( i "= j ), and of the off-diagonal elements in L , are related to the level of sectoral detail (that is, the number of sectors) in the model. We will return to this topic in Chapter 4 , when we consider the effects of aggregating (combining) sectors in an input–output model. (In Appendix 2.2 we examine the conditions under which a Leontief inverse matrix will always contain only non-negative elements, as logic suggests should always be the case.)
2.3.4 Numerical Example: The US 2003 Data We present a highly aggregated, seven-sector version of the 2003 US input–output coef- ficients matrix and its associated Leontief inverse in Tables 2.7 and 2.8. (Appendix B contains a series of such tables over time for the US economy at the seven-sector level of aggregation.) It is important to note that these data for the US represent domestically produced inputs; this requires explanation. Imports are generally divided into two categories: “competitive” and “non- competitive” imports (or “competing” and “non-competing”).
