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- The Problem
Suppose that we wished to predict the probable effects of a major cutback in, say, defence expenditure. From the simple macro-model, we know that, if this were accompanied by no other change in government policy, the result would be a reduction in income and employment. The simple macro-model also predicts that this reduction could be avoided by an offsetting increase in some other item of government expen- diture (thus keeping G constant), or an appropriate reduction in tax rates (increasing C) or, perhaps, a reduction in interest rates (increasing I). Do we in fact expect a reduction of £x million in expenditure on aircraft, electronic equipment, etc., matched by an increase of £x million on school and hospital building, to leave equilibrium happily undisturbed? We do not, for the simple reason that resources are not perfectly mobile. If we start from (more or less) full employment, we expect a bottleneck in the building industry and unemployment among aeronautical and electronics engineers, in the short run, at least. We should therefore notice that, in our simple macro-models, we have been implicitly assuming the existence of only one good. Input-output analysis recognizes the existence of many goods, and furthermore takes into account the fact that some goods are used to produce others. Thus our defence-school building problem is complicated by the fact that a reduction in aircraft building will lead to lower demand for aluminium and jet engines, while increased school and hospital building will lead to increased demand for timber, cement and plumbing materials. The increased demand for plumbing materials will in turn lead to increased demand for lead, copper, plastics and so on. Input-output analysis is designed for the quantitative study of these inter-industry relations. It is thus designed to answer practical questions about, for example, the effects of a switch in government spending of the sort we have been discussing, or the feasibility of a spending programme (where will the bottlenecks occur?), or the specific implications for industry of a projected plan. The answers all depend, however, on certain critical assumptions. The required data are also very expensive to collect. Let us now lay out the model.
1.1. Inter-Industry Accounts. Let us start by assuming a closed economy, and also suppose that the economy has been divided, perhaps by civil servant organizing a census of production, into four productive sectors: Services; Agriculture, Mining etc.; Public Utilities; and Manufacturing. We also assume that, from some source such as a census of production, we have the following information for some time period such as a year: for each industry, the value of its sales to each industry (sales of intermediate products) and to final consumption; and, for each industry, the value of its purchases from each industry (intermediate products) and its value added. We can arrange this information in a transaction matrix such as Table 1. In the transaction matrix, the row for each industry gives the amounts it supplies to other industries and the column its inputs purchased from other industries. The characteristic element, which we might write Xij , is thus the value of the output of the i th^ industry used up as an intermediate input by the j th^ industry. Let us look at some of these elements. X 12 is the value of the purchases from the Services industry by Agriculture, etc. It is thus composed of items such as insurance premiums, banking and legal charges, consulting fees and the like. X 13 and X 14 carrying the same interpretation. What about
1
Table 1. A 4-Sector Transaction Matrix (in $’s)
1 2 3 4 Zi = ∑ j Xij Yi Xi = Zi + Yi
- Services X 11 X 12 X 13 X 14 Z 1 Y 1 X 1
- Agriculture, Mining, etc. X 21 X 22 X 23 X 24 Z 2 Y 2 X 2
- Public Utilities X 31 X 32 X 33 X 34 Z 3 Y 3 X 3
- Manufacturing X 41 X 42 X 43 X 44 Z 4 Y 4 X 4 Vi = Wi + Ri V 1 V 2 V 3 V 4 V = Y
the elements on the principal diagonal, such as X 11 , X 22? X 11 is the value of the output of the Services industry used up by the Services industry itself. It is thus composed of items such as insurance premiums and legal fees paid by banks, bank charges and legal fees paid by insurance companies, etc. Similarly, X 22 is composed of items such as the value of grain and root crops fed to animals. The elements on the principal diagonal thus give the values of the sector’s consumption of its own output. We could, in fact, net out this consumption before presenting the transaction matrix (and they are sometimes built net), but we shall continue to use the gross form. There are many matters here requiring comment, but let us first tidy up the accounts. Suppse that we add up the elements of a row, forming the sum
Zi =
j
Xij = Xi 1 + Xi 2 + Xi 3 + Xi 4 (1)
Zi is the total value of the output of the ith^ industry used up in the productive process, that is, not available for final demand. We denote final demand by Yi, and we write Y for the sum, ∑ i Yi. In our one-good macro-models, we have Y = C + I + G (2)
This gives us our definition of final demand: the value of output that goes to households, to expanding productive capacity and to the public sector. It is the business of the macro-models to explain the determination of the total, Y. Here, we are interested in its industrial composition. If we assume that this is known, we have the column Yi. We now form the sums
Xi = Zi + Yi (for all i) (3)
and Xi is obviously the total value of output of the ith^ industry. Equation (3) (there are as many of them as there are sectors) are accounting identities, sometimes known as balance equations. Let us write beneath each column of the matrix the value added in each industry. This is composed of the wage bill, Wi, and the return to capital, Ri. If we take the sum
V =
i
Vi =
i
(Wi + Ri) (4)
we have national income, where V = Y. (Remember that R is a residual, so the books must balance.) Finally, suppose that we sum the elements of a column, and add the value added as well. We must have ∑ i
Xij + Vj = Xj (5)
In other words, the total value of the output of the j th^ industry must be equal to the value of its intermediate inputs, its wages bill and what is left over (R), if anything. We note here that it is only possible to sum the columns if we work in value terms. If we drew up a transaction matrix in physical units, row sums would make sense. If, however, we then attempted to sum columns, we would be trying to add, for example, units of banking services to bushels of wheat, which is impossible. A column sum,
(i) Each element must be non-negative: we rule out negative inputs. (ii) No element can exceed unity: we rule out negative outputs. If some element exceeded unity, it would mean, for example, that the value of the coke used in making a ton of steel exceeded the value of the steel. We assume that such an activity would not be pursued. (iii) The sum of the elements in each column must be less than unity. That is,
i aij^ <^ 1.^ If this were not true, it would mean that the total value of intermediate products used by an industry exceeded the value of its output. This in turn would mean that the value added by that industry was negative. Now, this is not impossible, but, if we assume that the wage bill cannot be negative, it means that the industry must be making losses (indeed, losses greater in absolute value than its wage bill). An industry in which value added is negative is not covering variable costs (intermediate inputs plus the wages bill), and we know from elementary micro theory that in such a case losses will be reduced by closing down. Thus we do not want to describe such an industry in our technology matrix at all. (iv) We have already noticed that we have built in the assumption of constant returns to scale: otherwise, we could not describe the technology by a constant-coefficient matrix. (v) We should notice that we have also built in the assumption that there are no externalities. An externality in production would exist if, for example, a factory discharged waste into a river so that a factory farther downstream had to use resources to clean the water before it could use it. In this case, the resource requirement of the second factory would not depend solely on its outputs but would also depend on the activity of the first factory. If we donote the two outputs by q 1 and q 2 , we should have to write the production function for q 2 in the form q 2 = f (q 1 , X 1 , X 2 ,... , Xn) (where the X’s are the inputs). The presence of q 1 as an argument of this function would be sufficient to prevent us describing resource requirements by (7).
- The ‘Leontief Matrix’
Let us rearrange (8). Subtracting Ax from both sides, we have
x − Ax = y (9)
We now factor out x to obtain (I − A)x = y (10) where I is the 4 by 4 unit matrix. The matrix I − A is often known as the ‘Leontief matrix’, after Wassily Leontief, the originator of input-output analysis.^1 We should see what it looks like:
I − A =
1 − a 11 −a 12 −a 13 −a 14 −a 21 1 − a 22 −a 23 −a 24 −a 31 −a 32 1 − a 33 −a 34 −a 41 −a 42 −a 43 1 − a 44
It is thus a matrix with positive numbers (fractions) along the principal diagonal and negative fractions elsewhere. Finally, we suppose that we are given the components of the vector y, final demand. That is, we have the industrial composition of C + I + G, often known as the ‘final bill of goods’. This is the list of required hospitals, airplanes, etc. that we discussed earlier, plus the commodity breakdown of planned or expected consumption and investment. Thus the problem is, given the coefficient matrix
(^1) see W. Leontief, Structure of the American Economy, 1919-1939: an empirical application of equlibrium analysis, 2nd ed. (London 1951). Professor Leontief was awarded the Nobel Prize in 1973 largely for his work in this field.
A, and the final bill of goods, y, find the outputs required of each industry, that is, the vector x. From (10), we see that the problem is solved by
x = (I − A)−^1 y (12)
Thus, given A, all we have to do is form the Leontief matrix (I − A) and obtain its inverse. We can then find required outputs for any final bill of goods merely by multiplying y by x = (I − A)−^1. If, in addition, we know each industry’s requirements of labour, then we can find the vector of jobs that goes with a final bill of goods. It turns out, in fact, that finding the inverse may be more conveniently done by approximation methods than by the method of the adjoint matrix.
